UC-NRLF 


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i«0?^  ^INFINITESIMAL  ^LCULUS— FISHEF 


1      UN 


BERKELIY 

BRARY 


UNIVERSITY  OF 
V       CALIFORNIA 


I 


Digitized  by  the  Internet  Arciiive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/briefintroductioOOfishrich 


A   BRIEF    INTRODUCTION 


TO 


THE    INFINITESIMAL   CALCULUS 


j2^^ 


A  BRIEF  INTRODUCTION 


TO  THE 


Infinitesimal  Calculus 


DESIGNED  ESPECIALLY  TO  AID   IN  READING 

MATHEMATICAL  ECONOMICS  AND 

STATISTICS 


BY 


IRVING   FISHER,  Ph.D. 

Professor  of  Political  Economy  in  Yale  University 
Co-author  op-  Phillips's  and  Fisher's  "  Elements  of  Geometry' 


THIRD  EDITION 


THE   MACMILLAN   COMPANY 

LONDON :  MACMILLAN  &  CO.,  Lxa 
I92I 

All  rights  reserved 


Copyright,  1897, 
By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped  1897.      Reprinted  April,  igooi 
July,  1901;  February,  1904;  March,  1906;  March,  1909; 
September,  1916. 


Notisooti  l^rees 

1.  8.  Gushing  &  Co.  —  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


Add'l 


0^505 


PREFACE 


This  little  volume  contains  the  substance  of  lectures  by 
which  I  have  been  accustomed  to  introduce  the  more 
advanced  of  my  students  to  a  course  in  modern  economic 
theory.  I  could  find  no  text-book  sufficiently  brief  for  my 
purpose,  nor  one  which  distributed  the  emphasis  in  the 
desired  manner.  My  object,  however,  in  preparing  my 
notes  for  publication  has  not  been  principally  to  provide  a 
book  for  classroom  use.  It  must  be  admitted  that  very  few 
teachers  of  Economics  as  yet  desire  to  address  their  stu- 
dents in  the  mathematical  tongue.  I  have  had  in  mind  not 
so  much  the  classroom  as  the  study.  Teachers  and  students 
alike,  however  little  they  care  about  the  mathematical 
medium  for  their  own  ideas,  are  growing  to  feel  the  need  of 
it  in  order  to  understand  the  ideas  of  others.  I  have  fre- 
quently received  inquiries,  as  doubtless  have  other  teachers, 
for  some  book  which  would  enable  a  person  without  special 
mathematical  training  or  aptitude  to  understand  the  works 
of  Jevons,  Walras,  Marshall,  or  Pareto,  or  the  mathematical 
articles  constantly  appearing  in  the  Economic  Journal^  the 
Journal  of  the  Royal  Statistical  Society,  the  Giornale  degli 
Economisti,  and  elsewhere.  It  is  such  a  book  that  I  have 
tried  to  write. 

S81 


vi  PREFACE 

The  immediate  occasion  for  its  publication  is  the  appear- 
ance in  English  of  Cournot's  Principes  mathematiques  de  la 
theorie  des  richesses^  in  Professor  Ashley's  series  of  "  Eco- 
nomic Classics."  The  "  non- mathematical  "  reader  can 
only  expect  to  understand  the  general  trend  of  reasoning  in 
this  masterly  Httle  memoir.  If  he  finds  it  as  stimulating  as 
most  readers  have,  he  will  want  to  comprehend  its  notation 
and  processes  in  detail. 

I  have  tried  in  some  measure  to  meet  the  varying  needs 
of  different  readers  by  using  two  sorts  of  type.  If  desired, 
most  of  the  fine  print  may  be  omitted  on  first  reading,  and 
all  on  second.  The  reader  is,  however,  advised  not  to  pass 
over  all  of  the  examples. 

Although  intended  primarily  for  economic  students,  the 
book  is  equally  adapted  to  the  use  of  those  who  wish  a  short 
course  in  "  The  Calculus  "  as  a  matter  of  general  education. 
I  therefore  venture  the  hope  that  teachers  of  mathematics 
may  find  it  useful  as  a  text-book  in  courses  planned  espe- 
cially for  the  "  general  student."  I  have  long  been  of  the 
opinion  that  the  fundamental  conceptions  and  processes  of 
the  Infinitesimal  Calculus  are  of  greater  educational  value 
than  those  of  Analytical  Geometry  or  Trigonometry,  which 
at  present  find  a  conspicuous  place  in  our  school  and  college 
curricula.  Moreover,  they  are  almost  as  easily  learned,  and 
far  less  easily  forgotten. 

IRVING.  FISHER. 

New  Haven,  September,  1897. 


PREFACE  TO  THE  THIRD  EDITION 

In  the  present  edition  have  been  incorporated  several 
changes  and  additions  originally  prepared  for  the  German 
translation  of  1904  and  for  a  Japanese  translation  in  prep- 
aration. 

A  preliminary  statement  of  the  concepts  of  limits  and 
several  new  examples  have  also  been  inserted. 

IRVING  FISHER. 
November,  1905. 


CONTENTS 

PAGB 

Introduction xi 

CHAPTER  I 
The  General  Method  of  Differentiation   .        .       ,       ,        i 

CHAPTER  II 
General  Theorems  of  Differentiation        .        .        .        .      i6 

CHAPTER  III 
Differentiation  of  the  Elementary  Functions  ...      30 

CHAPTER  IV 
Successive  Differentiation  —  Maxima  and  Minima     .        .      37 

CHAPTER  V 
Taylor's  Theorem 49 

CHAPTER  VI 
Integral  Calculus 57 

APPENDIX 
Functions  of  More  than  One  Variable       .       .       .       .73 


INTRODUCTION 

The  reader  of  the  following  book  should  be  familiar  with 
ordinary  algebraic  operations  and  with  the  concepts  of  vari- 
ation and  limits,  a  brief  statement  of  which  is  here  appended. 

Continuous  Variation.  —  Suppose  the  line  ab  to  represent 
all  possible  magnitudes  between  —  a  and  -f  b ;  suppose  om 
to  represent  one  magnitude  between  —  a  and  +  b ;  this 
magnitude  is  said  to  vary  continuously  when  it  increases  or 

—-a                                  o                  m                 m\      nity     mo             +  d 
Z i L, I I 1 

Fig.  I. 

decreases  in  such  a  manner  that  m  may  occupy  any  position 
whatever  between  —  a  and  -f  b. 

Limits.  —  If  we  conceive  om  to  have  an  infinite  succes^on 
of  magnitudes  such  that  ///  may  occupy  the  positions  nii,  Wj, 
Wg,  etc.,  making  the  ultimate  difference  between  ob  and  om 
less  than  any  assignable  positive  quantity,  then  om  is  a  vari- 
able and  ob  is  its  limit 

It  is  clear,  then,  that  the  difference  between  the  limit  ob 
and  the  variable  om  is  another  variable  magnitude  whose 
limit  is  zero.  A  variable,  with  a  limit  zero,  is  called  an 
infinitesimal. 


xii  INTRODUCTION 

Application  to  Infinite  Series.  —  In  a  converging  infinite 
series,  the  sum  of  each  successive  term  and  those  preceding 
approaches  a  magnitude  understood  to  be  designated  by 
the  series.     This  magnitude  is  called  the  *  sum '  of  the  series. 

Thus,  the  repeating  decimal  .666  •••, 

ID        lO^        lO^        ID* 

means  a  series  of  successive  magnitudes,  viz.: 


{a)  — ,  which  is  less  than  |. 

{b) 1 2>  which  is  less  than  |,  but  more  nearly  approxi- 
mates I  than  {a), 

(c) 1 sH 5,  which  is  less  than  |,  but  more  nearly 

approximates  |  than  (^). 

(d) 1 r.  H 7,  H 7,  which  is  less  than  |,  but  more 

^    '    lO       lo-^       lO^       lO^  ^' 

nearly  approximates  |  than  {c). 

Thus,  as  the  number  of  terms  of  the  series  is  increased, 
the  sum  of  the  terms  remains  always  less  than  |,  but  approx- 
imates ultimately  as  nearly  |  as  may  be  desired,  i.e.  converges 
towards  |.  We  therefore,  by  convention,  speak  of  J  as  the 
'  sum,'  or  limit,  of  this  infinite  series. 

Theorems. 

I.  The  limit  of  the  sum  of  two  different  variables  {which 
approach  limits)  is  the  sum  of  the  limits  of  those  variables. 


INTRODUCTION  xiX 

2.  The  limit  of  the  difference  of  two  different  variables 
{which  approach  limits)  is  the  difference  of  the  limits  of  those 
variables. 

3.  The  limit  of  the  product  of  two  different  variables 
{which  approach  limits)  is  the  product  of  the  limits  of  those 
variables. 

4.  The  limit  of  the  quotient  of  two  different  variables 
{which  approach  limits)  is  the  quotient  of  the  limits  oj  those 
variables. 


INFINITESIMAL   CALCULUS 

CHAPTER   I 

THE   GENERAL   METHOD   OF   DIFFERENTIATION 

1.  The  Infinitesimal  Calculus  treats  of  the  ultimate  ratios 
of  vanishing  quantities.  This  definition,  however,  can  only 
become  intelligible  after  some  actual  acquaintance  with 
"ultimate  ratios." 

2.  The  conception  of  a  limiting  or  ultimate  ratio  is  funda- 
mental in  many  familiar  relations.  It  is  impossible,  without 
it,  to  obtain  a  clear  notion  of  what  is  the  velocity  of  a  body 
at  an  instant.  The  average  velocity  of  the  body  during  a 
period  oi  time  may  readily  be  defined  as  the  quotient  of  the 
space  traversed  during  that  period  divided  by  the  time  of 
traversing  it.  If  a  steamer  crosses  the  Atlantic  (3000  miles) 
in  6  days,  we  may  say  that  the  average  speed  is  3000  -i-  6, 
or  500,  miles  per  day.  But  this  does  not  tell  us  the  speed  at 
various  points  in  the  voyage,  under  head  winds,  storms,  or 
other  conditions,  favorable  or  unfavorable.  What,  for 
instance,  was  the  speed  at  noon  of  the  third  day  out?  We 
may  obtain  a  first  approximation  to  the  desired  result  by 
taking  the  average  speed  for  a  short  time  after  the  given 
instant ;  that  is,  taking  the  ratio  of  the  distance  traversed 


2  INFINITESIMAL   CALCULUS 

during  (say)  the  following  hour  to  the  time  of  traversing  it, 
which  is  -^-^  of  a  day.  If  this  distance  be  20  miles,  we  obtain 
20  -7-  -Jj,  or  480  miles  per  day,  as  the  average  speed  during 
that  hour.  For  a  second  approximation  we  take  a  minute 
instead  of  an  hour ;  for  a  third,  a  second  instead  of  a  minute, 
and  so  on.  The  ratio  of  the  space  traversed  to  the  time  of 
traversing  it  becomes  closer  and  closer  to  the  true  speed. 
Though  both  the  time  and  space  approach  zero  as  limit, 
their  ratio  does  not.  The  limit  which  this  ratio  approaches, 
or  the  ultimate  ratio  of  the  distance  traversed  to  the  time  of 
traversing  it  when  both  distance  and  time  vanish,  is  the  pre- 
cise speed  at  the  instant. 

3.  Let  us  apply  this  method  of  obtaining  velocity  to 
bodies  falling  in  a  vacuum.  We  know  from  experience  that 
the  distance  fallen  equals  sixteen  times  the  square  of  the 
time  of  falling,  i.e.  s=  16/^,  where  s  is  the  distance  fallen 
from  rest  (measured  in  feet),  and  /  is  the  time  of  faUing  (in 
seconds).  Consider  the  body  at  some  particular  instant,  / 
being  the  time  to  this  particular  point  and  s  the  distance. 
Suppose  we  wait  until  the  time  has  increased  by  a  small 
increment  A/,  during  which  the  body  increases  its  distance 
from  the  starting-point,  s^  by  the  small  increment  Aj".  Since 
the  above  formula  holds  true  of  all  points,  it  holds  true  now, 
when  the  time  is  t-\-l^t,  and  the  distance  is  j-f-Aj.     That  is, 

s  +  As=  16  (t  +  Aty. 
This  gives 

j  +  Ai-=  16/2+32/'.  AZ-h  i6(My. 

But  s  =i6t\ 

Subtracting,  we  have 

A.f=32/.  A/+  i6(A/)2, 

li 


GENERAL   METHOD    OF  DIFFERENTIATION        3 

whence  — *^  =  3 2  ^  +  1 6  A/.  (i) 

This  is  the  average  velocity  during  the  small  interval  A/. 

Thus,  if  A/  =  I  second  and  /  be  5  seconds,  the  average  speed  of  the 
body  during  that  half  second  (viz.,  the  one  beginning  5  seconds  from 
rest)  is  32  X  5  +  1 6  X  ^,  or  1 68  feet  per  second.  If  we  take  ^\^  of  a  sec- 
ond instead  of  \,  we  have  32  x  5  +  16  X  y^^,  or  1 60.1  feet  per  second. 

Thus,  by  taking  A/  smaller  and  smaller,  we  obtain  the 

Ax 
average  velocity  —  for  a  smaller  and  smaller  interval  of  time 
A/ 

immediately  after  the  completion  of  the  fifth  second.     The 

Aj" 
limit  which  —  approaches,  as  A/  approaches  zero  as  its 
A/ 

limit,  is  called  the  velocity  at  the  very  instant  of  completing 

the  fifth  second. 

Its  value  is  exactly  160,  as  is  evident  from  the  right-hand 

member  of  equation  (i),  which  approaches  as  its  limit  (as  / 

is  5  and  A/  approaches  zero), 

32  X  5  +  16  X  o,  or  160. 

In  general,  to  express  the  limit  of  both  sides  of  equation 
(i)  when  A/  approaches  zero,  we  write 

lim  —  =  '?2  /. 
A/      ^ 

4.  The  student  will  observe  that,  as  A/  approaches  zero, 
Aj  also  approaches  zero,  since  a  body  cannot  pass  over  any 
distance  in  no  time.     He  must  be  warned:>  however,  against 

expressing  the  limit  of  —  by  -,  which,  of  course,-  is  quite 

indeterminate. 

But  in  spite  of  the  fact  that  the  ratio  of  these  limits  of  Ai 
and  A/  is  indeterminate,  the  limit  of  the  ratio  of  Ax  and  A/ 


4  INFINITESIMAL    CALCULUS 

may  be  entirely  determinate.     It  is  only  with  this  latter  con- 

ception,  viz.  the  limit  of  — ,  or  lim  — ,  that  the  student  has 

to  deal.  ^'  ^' 

The  Hmit  of  the  ratio  of  the  vanishing  quantities  Aj-  and 

Ai" 
A/,  or  lim  — ,  is  called  the  "  derivative  "  of  j  with  respect  to 

^*  As 

t ;  because,  from  s=  16  /^  we  denve  hm  —  =  -12  f. 

A/     ^ 

In  fact,  we  may  speak  of  either  member  of  the  latter  of 
these  two  equations  as  the  derivative  of  either  member  of  the 
former  equation.     For  instance,  32  /  is  the  derivative  of  16  /^. 

5.  Other  names  and  notations  are  also  used.  Thus  in- 
stead of  lim  —  it  is  usual  to  employ  the  shorter  symbol  — . 

A/  ^    ^  ^  dt 

In  this  expression  ds  and  dt  are  called  differentials  of  s  and  /, 
just  as  Ai"  and  A/  are  called  increments  of  s  and  /.  But  they 
are  not  zeros.  They  have  no  definite  value  individually.  We 
may  select  any  value  we  please  for  one  of  them.  But  when 
this  one  is  fixed,  the  other  is  also,  since  the  two  must  be  kept 

in  a  ratio  equal  to  lim  — .  We  say  therefore  that  the  differ- 
entials ds  and  dt  are  any  two  quantities  which  bear  to  each 
other  the  ratio  which  is  the  limit  of  the  ratio  between  Aj 
and  A/. 

Other  names  for  lim  —   or  —.besides  "derivative,"  are 
A/        dt' 

"  differential  quotient "  and  "  differential  coefficient." 

6.  In  the  particular  case  considered  above,  the  differ- 
ential quotient  is  a  velocity  and  may  be  denoted  by  v. 
Equation  (2)  thus  becomes*    v=T,2t. 

*  If  distance  be  measured  in  centimetres  instead  of  in  feet,  we  should 
have  V  =  980  /,  and  in  general  v  =gty  where  ^  is  a  constant  depending 
for  its  numerical  value  on  the  units  chosen  for  measuring  space  and  time. 


GENERAL  METHOD    OF  DIFFERENTIATION         5 

Velocity  at  a  point  may  now  be  defined  as  the  ultimate 
ratio  of  the  space  traversed  Just  after  passing  the  point  to  the 
time  of  traversing  it  when  the  space  and  time  approach  zero 
as  limit. 

7.  Examples. 

1.  What  is  the  velocity  of  a  body  which  has  fallen  10  seconds  ? 
100  seconds  ?   i^  seconds  ? 

2.  What  is  the  velocity  of  a  body  which  has  fallen  16  feet  ? 
Hint.  —  First  find  how  many  seconds  it  has  fallen  by  using  j=i6/-. 

3.  What  is  the  velocity  of  a  body  which  has  fallen  64  feet  ?  4  feet  ? 
I  foot  ?   2  feet  ? 

4.  It  being  known  that  a  body,  falling  not  from  rest,  but  with  an 
initial  velocity  of  5  feet  per  second,  obeys  the  law 

s=  16/2+5/,  (l) 

what  will  be  its  velocity  at  the  end  of  any  time  t  ? 

Hint.  —  Let  /  receive  an  increment  A/,  causing  s  to  increase  by  ^s^ 

so  that 

s-\-£^s=  l6(/  +  A/)2  +  5(/  +  A/).  (2) 

Subtract  (i)  from  (2),  divide  by  A/  and  then  reduce  A/  and  \s  to  zero. 

Ac 
Ans.  lim   -    =  ^2/  4-  C. 
A/      "^  ^ 

5.  What  will  be  the  velocity  at  the  end  of  10  seconds?  At  the  end 
of  69  feet  ? 

6.  It  being  known  that  a  body  falling  with  an  initial  velocitv  of  u 
obeys  the  law  s  =  \gt'^  +  m^i  what  will  be  its  velocity  at  the  end  of 
time/?     When/=3?    . 

8.  When  one  quantity  depends  upon  another,  the  first  is 
said  to  be  a  function  of  the  second.  A  change  in  the  second 
is  in  general  accompanied  by  a  change  in  the  first.  In  each 
case  the  limits,  within  which  the  function  relation  exists 
should  be  specified. 


6  INFINITESIMAL   CALCULUS 

Thus  the  distance  a  body  falls  from  rest  is  a  function  of  the  time  of 
falling,  for  how  far  the  body  falls  depends  on  how  long  it  has  fallen;  - 
the  demand  for  an  article  is  a  function  of  its  price,  for  if  the  price 
changes  the  demand  changes;  \i  y  —  x^,  then  jj/  is  a  function  of  jr,  for 
a  variation  in  the  magnitude  of  x  necessitates  also  a  variation  in  the 
magnitude  of  y. 

9.  When  one  quantity  is  a  function  of  another,  the  latter 
is  called  the  independent  variable^  and  the  former  the  de- 
pendent variable. 

The  distinction  between  the  independent  and  the  depend- 
ent variable  is  only  for  convenience  of  expression.  The 
two  may  be  interchanged. 

Thus,  as  the  distance  of  a  falling  body  from  the  starting-point 
changes,  there  is  also  a  change  in  the  time  it  has  taken.  Hence  we 
may  say  that  "  time  of  falling  "  is  a  function  of  "  distance  fallen."  Simi- 
larly price  may  be  regarded  as  a  function  of  demand.  Again,  y  =  x'^ 
may  be  written  x  =  Vy,  thus  making  x  a  function  of  y.  The  idea  of 
functional  dependence  is  therefore  quite  different  from  that  of  causal 
dependence.     Functional  dependence  is  a  wm/m^/ relation. 

In  the  example  of  falling  bodies  s  was  a  function  of  /,  and 
what  we  accomplished  was  to  find  the  differential  quotient 
or  derivative  of  that  function.  The  derivative  in  this  case 
was  a  velocity.  In  general  the  process  of  finding  the  differ- 
ential quotient  of  any  given  function  is  called  differentiation^ 
and  is  the  subject  matter  of  the  Differential  Calculus,  one 
of  the  two  branches  into  which  the  Infinitesimal  Calculus  is 
divided.  The  Differential  Calculus  will  occupy  us  in  the 
first  five  chapters  of  this  book. 

10.  A  second  important  application  of  the  idea  of  a  differ- 
ential quotient  of  a  function  is  to  the  tangential  direction  of  a 
curve  at  any  point  on  it.  The  Calculus  enables  us  to  conceive 
in  the  most  general  manner  of  a  tangent  to  a  curve.     The 


GENERAL   METHOD    OF  DIFFERENTIATION         7 

Student  should  observe  that  the  usual  definition  of  a  tangent 
to  a  circle  will  not  apply  to  any  and  all  curves.  A  straight 
line  may  have  only  one  point  in  common  with  a  curve  and 
yet  cut  it  and  not  be  tangent. 

1 1 .    Let  RS  be  a  curve  whose  equation  is 

y=^\^-^x-x^,  (i) 

That  is,  for  atiy  point  P  upon  it,  the  "  ordinate,"  y  (or  dis- 
tnnrp^  P4  from  that  point  to  the  horizontal  axis),  is  related 


Fig.  I. 

to  the  **  ahsrj<;sa^''WW  ^l^gfonr^^^  Q^  from  the  vertical  axis), 
in  the  manner  expressed  by  (i).  PA  is  a  function  of  OA  ; 
i.e.  the  height,  PA,  of  any  point  P  on  the  curve  depends 
upon  its  distance,  OA,  from  the  vertical  axis. 

What  is  the  direction  of  the  curve  at  the  point  P  ?  The 
direction  from  the  point  P  to  another  point  P  is  the  direc- 
tion of  the  secant  line  QPP.     The  point  P  has  for  abscissa, 


8  INFINITESIMAL    CALCULUS 

X  4-  A^,  and  for  ordinate,  y  +  A>'.  Since  the  relation  (i) 
holds  true  of  all  points  on  the  curve,  it  holds  true  of  P . 

Hence        j  +  Ay  =  i  +  5  (jv  +  A^)  —  {x  -{-  Ajc)^, 

or  ^H- Aj' =1  +  5^  +  5  A^  —  ^— 2:vAj£:—  {J^ocf. 

Subtracting  y=.  i  -\-  ^x  —  0^, 

we  have  Aj  =  5  ^x  —  2  x  ^x  —  (Ajv)^, 

whence  —  =  k  —  2  x  —  Ajc. 

Ax      ^ 

We  may  pause  here  a  moment  to  see  what  this  result 

means.     ^^  or  —  is  the  "  slope  "  of  the  line  Q'FP'.    That 
Ax       PC  ^  ^ 

is,  it  is  the  rate  at  which  a  point  moving  from  Q'  toward  P 
rises  in  proportion  to  its  horizontal  progress.  It  is  the  same 
sort  of  magnitude  as  that  referred  to  as  the  "  grade  "  of  an 
uphill  road  which  rises  "  so  many  feet  to  the  mile  (hori- 
zontally)."    If    -^  =  — ,  QPP  rises  one  foot  in  every  ten 

horizontally.     The  "  slope  "  of  a  line  shows  its  direction. 

At' 
The  equation   -^  =  ^  —  2x  —  ^x  shows  that  the  "slope"  of  the 

secant  line  Q'PP'  is  to  be  found  by  taking  5  and  subtracting,  first, 
two  times  the  number  of  units  in  OA  and  then  the  number  of  units 
in  AB.     For  instance,  if  OA  =  2  and  AB  =  ^,  then 

-^=  5  —  2x2  —  A  =  *; 
t.g.  the  secant  slopes  i  foot  up  for  every  2  feet  sidewise. 

12.  But  we  have  not  yet  reached  the  tangent  at  P.  Let 
the  point  P'  be  gradually  shifted  along  the  curve  toward  P 
until  it  ultimately  coincides.     The  secant  QP'  will  gradually 


GENERAL  METHOD   OF  DIFFERENTIATION        9 

change  its  direction  and  approach  a  limiting  position  QP, 
This  limiting  position  we  call  the  tangent.     Its  slope  is 
dy 

Thus,  if  x(i.e.  OA)  is  2,  ^=  i.    That  is,  QP  is  inclined  at  45^ 

dv 
If  4;  is  4,  -^  =  —  3 ;   i.e.  the  curve  slopes  down^  not  up. 


Fig.  2.  —A,  positive  slope;  B,  zero  slope;  C,  negative  slope. 

Examples. 

1.  What  is  the  slope  of  the  tangent  to  the  above  curve  at  the  point 
whose  abscissa  is  i  ?  o  ?  2J  ?  What  does  the  answer  to  the  last 
mean  ?     3  ?     What  does  this  mean  ?     6  ?     —  i  ? 

2.  Derive  the  formula  for  the  slope  of  the  tangent  to  the  curve 
y  —  \  -\-  X  -\-  x^. 

13.  To  construct  a  tangent  at  P,  all  we  need  to  do  is  to 
draw  a  Une  through  /'with  the  required  slope.  Thus,  if  we 
wish  the  tangent  to  the  point  whose  abscissa  is  i,  we  find 
from  the  above  formula  that  its  slope  is  3.  We  therefore 
lay  off  a  horizontal  line  LM  (Fig.  i)  equal  to  any  length  ^jc, 
and  at  its  extremity  erect  a  vertical,  MN,  equal  to  three  times 
as  much,  or  dy.  Draw  LN \  this  has  the  required  direction. 
Then  through  P  draw  a  line  parallel  to  LN.  This  will  be 
the  tangent. 

We  may  also  call  PC,  dx  and  /"'C,  dy,  for,  by  Sec.  5,  dx 
and  dy  are  simply  any  two  magnitudes  having  a  ratio  equal 

to  the  limit  of  -^  when  A^  approaches  zero  as  its  limit. 
t^x 

The  prublem  of  drawing  a  tangent  and  calculating  its  slope  was  one 
of  the  chief  problems  which  gave  rise  to  .the  discovery  of  the  Calculus. 


10  INFINITESIMAL   CALCULUS 

14.  It  is  evident  that  we  could  approach  P  from  the  left  as  well 
as  from  the  right.  We  should,  however,  reach  the  same  limiting  posi- 
tion unless  there  should  be  an  angle  in  the  curve  at  the  point  /*  as  in 
Fig.  3.  In  this  case,  the  progressive  (/'A')  and  regressive  {HP)  tan- 
gents do  not  coincide. 

_^^_  'K 

?, 


Such  peculiar  points  are  not  considered  in  this  little  treatise.  All 
the  functions  are  such  that,  for  the  values  of  the  independent  variable 
which  are  considered,  the  progressive  and  regressive  derivatives  are 
identical.  The  curves  considered  are  all  "smooth,"  that  is,  have.no 
angles  or  sudden  changes  in  direction.  In  many  applications  of  the 
Calculus,  such  as  to  statistical  or  economic  diagrams,  it  is  often  con- 
venient first  to  smooth  out  the  curves  considered.  When  we  want  to 
see  from  a  plot  of  the  population  what  is  the  general  rate  of  increase, 
we  draw  a  tangent  not  to  the  plot  of  the  actual  figures,  but  to  a  smooth 
curve  coinciding  as  nearly  as  possible  with  the  plot. 

The  student  will  be  able  to  satisfy  himself  in  every  particular  case 
to  be  considered  that  the  progressive  and  regressive  derivatives  are 
identical. 

Thus,  for  j=i6/2  in  section  3,  let  /  receive  a  decrement  A't,  causing 
s  to  have  a  decrement  a's.     Then 

s  -  A's  =  i6(t  -  A'ty. 

Expanding,  subtracting,  and  dividing  as  before,  we  obtain 

^=32t-l6A't, 
A't 

which  reduces  at  the  limit  to 

d's 

—  =  32  /,  as  before. 

d't      ^ 

Indeed,  we  assume  in  general,  that  it  is  physically  impossible  for 
a  body  to  change  its  velocity  />er  saltum.     Hence  the  definition  of 


GENERAL  METHOD   OF  DIFFERENTIATION       11 

velocity  given  in  section  6  is  equivalent  to  the  following  alternative 
definition  :  the  ultimate  ratio  of  the  space  traversed  just  before  reaching 
the  point  to  the  time  of  traversing  it  when  the  space  and  time  ap- 
proach zero  as  limit. 

We  shall,  therefore,  henceforth  treat  only  of  functions  whose  deriva- 
tives are  continuous  and  which  are  themselves  continuous,  within  the 
limits  considered,  that  is,  which  in  changing  from  one  value  to  another, 
pass  continuously  through  all  intermediate  values. 

15,  We  have  seen  that  the  conception  of  an  ultimate  ratio 
clears  up  the  notion  of  velocity  in  mechanics  and  tangential 
slope  in  geometry.  It  is  also  applicable  to  much  else  in 
both  these  sciences  as  well  as  in  all  mathematical  sciences. 
Momentum,  acceleration,  force,  horsepower,  density,  curva- 
ture, marginal  utility,  marginal  cost,  elasticity  of  demand, 
birth  rate,  "  force  of  mortality,"  are  all  examples. 

The  conception  of  an  ultimate  ratio  or  of  the  derivative  of 
a  function  is  not  dependent,  however,  on  any  special  appHca- 
tion.     It  is  purely  an  abstract  idea  of  number. 

16.  Thus  let  two  variables  x  and  y  fulfil  the  equation 

;;  =  x^, 

where   «  is  a  constant  and  a  positive  integer.     We  may 

obtain  the  differential  quotient  -^  for   any  particular  value 

dx 
of  X,  as  follows  : 

Let  X  receive  an  increment  Ajc  producing  an  increment  of 

y  denoted  by  Ay.     Then,  by  the  binomial  theorem, 

^  +  Aj  =  (^  -h  Ajc)", 

2 
=  Jt:**  4-  WJC""^  A^  -I-  Ajx:^ (•••). 


12  INFINITESIMAL    CALCULUS 


Subtracting 

y  =  x\ 

we  have 

A);  =  «ji:"^AxH-(A^)2(...) 

Whence 

^=«:c«-^  +  A^  (...), 

ivhere  the  parenthesis  is  evidently  a  finite  quantity  and  re- 
mains finite  after  Ax  becomes  zero.  Hence,  when  Ajc: 
becomes  zero,  the  term  Ax(---)  becomes  zero,  and  the 
equation  becomes, 

dx 
17.  This  is  the  first  and  most  important  specific  formula 
which  we  have  reached  for  the  derivative  of  a  function.  It 
states  that,  to  obtain  the  derivative  of  .r",  a  power  oi  x,  we 
need  only  reduce  the  exponent  by  unity  and  use  the  old 
exponent  for  coefficient. 

Thus  the  derivative  of  x^  is  t^x^.  When  x  passes  through  the  value 
2,  3^2  becomes  12;   that  is,  j,  or  x^,  is  increasing  I2  times  as  fast  as  x. 

-^  is  the  rate  at  which  j  increases  compared  with  the  rate  we  make  x 
dx 

increase.  If  y  denotes  the  distance  of  a  moving  body  from  the  start- 
ing-point, and  X  denotes  the  time  it  has  moved,  y-,  or  3  x'^,  expresses 

its  velocity.  Again,  if  x  and  ^  are  the  "coordinates"  {i.e.  the  "ab- 
scissa" and  "ordinate")  of  a  curve  whose  equation  is  ^  =  ;f^,  then 
3  x'^  is  its  slope  at  the  point  whose  abscissa  is  x. 

Although  it  is  logically  unnecessary,  it  is  practically  helpful  to  pict- 
ure the  differential  quotient  as  a  possible  velocity  or  a  possible  slope. 
Of  the  two  independent  discoverers  of  the  Calculus,  Newton  seemed 
to  have  employed  the  former  image,  and  Leibnitz  the  latter.  New- 
ton's term  for  a  differential  quotient  was  "  fluxion." 

Examples.  —  1.  Find  the  derivatives  of  ^^^^  ^^  ^2^  ^^  What  is  the 
meaning  of  the  answer  to  the  last  ? 

2.  How  many  times  as  fast  does  y  increase  as  x  when  y  —x^  and 
A-  is  2  ? 

3.  How  fast  does  x^  increase  compared  with  x  when  ^  is  -  i  ? 
What  dofs  the  negative  answer  mean  ? 


GENERAL  METHOD    OF  DIFFERENTIATION       13 

1 8.  The  process  employed  in  this  chapter  for  obtaining 
the  derivative  of  a  function  is  called  the  *'  general  method  of 
differentiation."  It  consists  (i)  in  giving  to  the  independent 
variable  a  small  increment,  thus  causing  another  small  incre- 
ment* in  the  dependent  variable  or  function  ;  (2)  in  writing 
the  relation  between  the  two  variables  first  without  and  then 
with  these  increments  and  subtracting  the  first  from  the 
second ;  (3)  in  dividing  through  by  the  increment  of  the  in- 
dependent variable  ;   (4)  in  passing  over  from  -^  to  ^. 

This  process  should  be  thoroughly  mastered  by  the 
student,  for  it  contains,  in  embryo,  the  whole  of  the  Infini- 
tesimal Calculus. 

He  will  observe  that  the  order  of  steps  (3)  and  (4)  cannot 
be  inverted  without  producing  the  barren  result  0  =  0. 

19.  Nevertheless,  we  can  anticipate  the  result  of  step  (4) 
without  changing  from  the  form  of  (2).     Thus,  the  equation 

yields  at  step  (2)  : 

A>'=  2  A^-i-6 ;c  Ajt:-f-3 {J^xf-\- 15  x^ lx-\-\%x{p.xf-\-<^{p.xf 
^(2  +  6^4-15  ^')^^'  +(3  +  15  ^)(A^)'  +  5  (M'- 
It  can  readily  be  foreseen  that  step  (3)  {i.e.  dividing  by 
^x)  will  remove  the  first  \x,  and  reduce  the  exponents  of 
the  powers  of  Ajc  by  one,  and  that  therefore  when  step  (4) 
is  performed  {i.e.  reducing  \x  to  zero),  all  terms  beyond 
the  first  will  disappear,  leaving  2  +  6  jic  +  15 -^  ^s  the 
derivative.  Now  it  is  clear  that  this  result  could  have  been 
anticipated  simply  by  neglecting  the  terms  involving  powers 

*  Decrements  may  always  be  regarded  as  negative  increments. 


14  INFINITESIMAL    CALCULUS 

of  Ajc  higher  than  the  first,  and  taking  the  coefficient  of  the 
first  power  as  the  required  derivative. 

Though  this  process  of  neglecting  certain  terms  at  step 
(2)  is  a  mere  anticipation  of  what  must  necessarily  happen 
at  step  (4),  it  may  be  shown  to  be  perfectly  natural  in  situ. 
If  A^  be  less  than  one,  (A^)-  will  be  less  than  Ajc,  and 
(A^)^  less  than  (Ajt:)^,  etc.  By  making  A^  smaller  and 
smaller,  the  higher  powers  (Ajc)^,  (A^)^,  etc.,  can  be  made 
indefinitely  small,  not  only  absolutely,  but  in  comparison 
with  Ajc.  The  higher  powers  of  AJt  thus  growing  negligible 
relatively  to  Ajc,  the  terms  in  which  those  powers  occur  as 
factors  must  also  grow  negligible  (provided,  of  course,  the 
other  factor  composing  each  such  term  does  not  approach 
infinity  as  limit). 

Thus,  if  A^  is  y^^,  {Lxf  is  jir^,  and  (A;r)8  only  xinnjW*  Con- 
sequently in  the  equation 

Ay  =  (2  +  6  ^  +  15  0:2)  A;^ -j- (3  +  15  ;r)  (A^)2  +  5(A;r)3, 

we  can,  by  reducing  t^^  sufficiently,  make  the  terms  beyond  the  first 
as  small  as  we  please  compared  zuith  the  firsts  no  matter  what  be  the 
value  of  x^  so  long  as  it  is  finite,  thus  keeping  the  parentheses  finite. 
For  instance,  if  x  be  2,  we  have  A^  =  74  A;r  +  33 (A^)'-^  +  5(Ajr)3, 
Then,  if 

Lx  be  .01,  this  becomes 

Ar  =  .74  +  .0033  +  .000,005. 

If  A^  =  .001,  it  becomes 

t^y  -  .074  +  .000,033  +  .000,000,005. 
If  A^  =  .000,001,  it  becomes 

Aj  =  .000,074  +  .000,000,000,033  +  ,000,000,000,000,000,005, 

and  the  smaller  we  make  A;r,  the  more  negligible  become  the  terms 
involving  (A;r)2  and  (Ajf)^,  until  at  the  limit  they  become,  not  simply 
negligible  "  for  practical  purposes,"  but  absolutely  negligible. 


GENERAL  METHOD   OF  DIFFERENTIATION       15 

The  anticipatory  neglect  of  terms  involving  powers  of  ^x 
higher  than  the  first  often  saves  a  great  deal  of  labor. 

Examples. 

1.  Find  -^  when  y  =  j^, 

dx 

2.  Find  ^  when  v  =  ;r7  +  8;t«  +  4. 

dx 

3.  Find  ^  when  y  —  \ox^^, 

dx  ^ 

4.  Find  -^  when  y  =  ax"^  +  dx^,  tn  and  n  being  constant  and 
integral.  ^  Ans.  amx^-^  +  dnx"-^. 

5.  If  X,  the  side  of  a  square,  has  an  increment  ?,  what  will  be  the 
increment  of  the  area  of  the  square  ? 

6.  In  the  function  y  =  ;^  x"^  +  2  x,  find  the  value  of  x  when  y  in- 
creases 20  times  as  fast  as  x.  Ans.  x  =  3. 

Differentiate  the  following  functions : 

7.  jj/  =  3  al^x^  +  c. 

8.  y  =  4x^  —  yx^-\-2x^2a.  Ans.  20  x^  —  21  x"^ -\-  2. 

9.  y  =  ^j(^-{a  +  b')x, 

10.  yz^^b-^-xY-  bx\  Ans.  z  ^ -^  A  bx  ■\- z  x^. 


16  INI^INITESIMAL    CALCULUS 


CHAPTER   II 

GENERAL  THEOREMS   OF   DIFFERENTUTION 

20.   If  we  differentiate 

y—  2X 
by  the  general  method,  we  obtain 

Clearing  this  equation  of  fractions,  we  have 

dy=2dx.  (2) 

This  last  equation  is  simply  another  form  of  the  first,  and 
more  convenient  for  some  purposes. 

Thus,  dy  —  d  xdx  is  a  transformation  of 

dx 

which  in  turn  means  lim  — ^  =  6  ;r. 

^x 

6  ;f  is  a  differential  quotient  and  6  xdx  is  a  differential. 

These  conceptions  are  strictly  correlative.  To  obtain  the  differen- 
tial quotient  from  the  differential,  we  simply  divide  by  dx ;  to  obtain 
the  reverse,  we  multiply  by  dx. 


GENERAL    THEOREMS   OF  DIFFERENTIATION     Vi 

Examples. 

1.  What  is  the  differential  of  ;«^? 

2.  The  differential  quotients  of  ^r",  x^"^,  x*? 

21.  To  express  the  mere  fact  that  j  is  a  function  of  x, 
without  specifying  exactly  w/iaif  function,  it  is  customary  to 
use  the  letters  ^,  /,  <^,  if/  (and  rarely  others)  followed  by  x 
in  a  parenthesis.     They  may  be  regarded  simply  as  abbrevia 
tions  of  the  word  "  function."     Thus 

j  =  Function  of  jc 

is  abbreviated  to  y  =  J^(x) . 

It  is  to  be  observed  that  the  letters  F,/,  0,  ^,  etc.,  do  not  repre- 
sent quantities  like  x  and  ^,  but,  like  A  and  £^,  represent  operations 
on  quantities. 

22.  The  general  expression  for  a  function,  such  as  <^(^), 
is  often  used  to  express,  within  brief  compass,  any  special 
function.    Thus  if  we  have  the  equation 

1+^-6^  +  — 
^~      ^x        2  —x^  * 

we  may  shorten  this  to  y=<l>{x)  by  denoting  the  clumsy 
right-hand  member  by  <f}{x). 

Again,  if  we  have  a  definite  curve,  such  as  a  statistical 
diagram,  whose  coordinates  we  call  x  and  y,  we  may  use 

y=/{x) 

to  express  the  fact  that  y  is  related  to  x  in  the  particular 
manner  delineated  by  the  curve. 


IS  INFINITESIMAL    CALCULUS 

23.  The  differential  quotient,  or  derivative  of  a  function 
of  AT,  is  itself  a  function  of  x. 

To  denote  the  differential  quotient  of 

we  use  the  expression        F\oc). 

Thus  let  ^{x)  stand  for  ofi. 
Then  <^\x)  stands  for  (ix>. 

The  differential  oi  F{x)  is  therefore  expressed  by 
F\x)dx. 

24.  Another  meth'  d  of  expressing  the  differential  quo- 
tient of 

F(x) 

connects  it  with  the  general  method  of  differentiation.    Thus, 
if  X  receives  an  increment  A^,  Fix)  will  become 

F{x-\-^x), 

This  differs  from  its  original  value  F{pc)  by 

F{x^t^x)-F{x)'. 

The  ratio  of  this  increment  of  the  function  to  the  incre- 
ment ^x^  of  the  independent  variable  x^  is 

F{x-\-  ^x)  —  F(x) 

t^x 

-     i.    .,     .           r     F(x-^b.x^-  Fix) 
Its  hmit,  VIZ.       hm  — ^ — ^ ^-^, 

is  the  differential  quotient  of  Fipc)  ;  i.e.  is 

F\x). 

The  above  process  is  identical  with  the  general  method  of  differen- 
tiation, though  we  have  expressed  it  without  the  use  of  y.  We  might 
have  proceeded  as  follows : 


GENERAL    THEOREMS   OF  DIFFERENTIATION     19 
Put  F^pc)  equal  to  y  so  that   . 

y  =  ^W. 

Subtract  this  from  y  -}-  Ay  =  F(j:  -{^•^Ax),  '  ^ 

and  divide  by  Ax^  giving  \ 

Ay  _  F{x  +  Lx^-r  E{x) 
Ax~  Ax 

or,  at  the  limit 

±  =  lim  ^(^  +  ^^)-^(^) 
dx  Ax 

25.  Yet  one  more  notation  should  b'  .miliarized. 
Rather  it  is  a  new  application  of   ^n  old  one.      Instead 

of  writing  -^,  we  may  replace  y  ^   this   expression  by 

F(x)y  so  that  it  reads 

'iJZMl. 
dx 

The  student  will  do  well  now  to  release  his  mind  from  y  as  any 
necessary  element  in  the  analysis.  It  is  to  be  regarded  merely  as  a 
further  abbreviation  of  F(x). 

F{x)  rather  than/  is  to  be  thought  of  as  primarily  the  function  of 
(jf).  Thus,  in  our  introductory  example,  instead  of  denoting  space  by 
s  and  writing  s  =  16  i^,  we  need  only  say  if  /  denotes  time,  the  function 
of  /,  16 /2^  will  denote  space. 

So  also  if  X  denotes  the  abscissa  of  a  curve,  F{x)  instead  of  y  de- 
notes its  ordinate. 

Thus,  ^^f!)is2A:, 

c/x 

or  d(x^)=2xdx. 

Examples.-  ^^  =  ^        ^(^)=? 

We  thus  have  five  methods  of  denoting  the  differential 
quotient  ofy,  or  its  equal  F{x)  ;  viz. : 


20  INFINITESIMAL    CALCULUS 

26.   If  a  function  of  x  is  the  sum  of  several  functions  of 
X  i.€,  if 

then,  since  this  equation  holds  true  of  all  values  of  x,  it 
holds  true  when  x  becomes  x  +  Ajc,  so  that 

F{x  +  A^)  =/i<^  +  A^)  4-/2(^  +  A^)  +  -. 

Subtracting  the  upper  equation  from  the  lower,  and  divid- 
ing by  A^,  we  obtain 

Fix  +  Ay)  -  F{x)  ^  /i  (^  +  A^)  -/i  {x) 
Aa:  ^x 

^  f,{x-^^x)-Mx)  ^ 

Ax 

Now  let  Ax  approach  zero  as  its  limit.     Then  for  the 
limits  of  the  terms  in  the  above  equation,  we  have  : 

Ax  Ax 

or    .  F'(x)=/,'(x)-\-/J(x)+,etc.' 

That  is,  fke  differential  quotient  of  the  sum  of  several  func- 
tions is  the  sum  of  the  differential  quotients  of  those  functions. 
The  same  reasoning  establishes  the  corresponding  theorem 
for  the  difference  of  functions. 

Thus  the  differential  quotient  of  ^r^  _}.  ^  \^  ix-\-  ^x^. 
Sometimes  the  theorem  is  used  in  the  differential  form 

F'(x)c/x=/i'(x)dx+/2'ixyx+-, 

or  again  F'^xyx  =  [/i'(^)  4-/2' W  +  .••]  dx. 

Examples.  —  Find  the  differential  quotient  of: 

1.  x^  +  x^-X^.  2.   xT-x^  +  x.  3.    -x^-^x^^ 


GENERAL    THEOREMS   OF  DIFFERENTIATION      21 

27.  If  a  function  of  x  is  the  sum  of  another  function  of 
X  and  a  constant  quantity,  i.e.  if 

.    F{x)=f{x)^K,  (i) 

where  ^  is  a  constant,  then 

•F\x)=f{x\  (2) 

the  same  result  as  if  K  were  not  present  in  (i)  at  all.  The 
proof  of  (2)  is  simple.  When  x  becomes  x^i^x,  (i)  be- 
comes 

F{x  +  A^)  =/(^  +  Ax)  4-  K.  (i)' 

When  we  subtract  (i)  from  (i)',  K  disappears  entirely,  and 
we  have,  after  dividing  by  A„t, 

F{x  +  Ajc)  -  F{x)  _  fix  +  Ax)  -f(x) 
Ax  ""  Ax  ' 

which  reduces  at  the  limit  to  (2).  The  same  result  would 
be  obtained  if  in  (i)  K  were  preceded  by  the  minus  instead 
of  the  plus  sign. 

Hence,  to  obtain  the  derivative  of  the  sum  (or  difference) 
of  a  series  of  terms,  some  of  which  are  constants,  we  simply 
take  the  sum  (or  difference)  of  the  derivatives  of  all  the 
terms  which  are  functions  of  x,  ignoring  those  which  are 
constant. 

Thus,  if  j^  =  ^3  +  5^  ^  =  3;f2. 

Again,  the  derivative  of 

x^-x*  +  x  +  a-d-S  is  s^-4^^+i' 

The  foregoing  result  is  sometimes  expressed  by  regarding 
all  the  terms,  even  the  constants,  as  functions  of  x,  and 
saying  that  the  derivative  of  a  constant  term  is  zero. 


22  INFINITESIMAL    CALCULUS 

Examples.  —  Find  the  differential  quotient  of: 
I.  x^  +  2.  2.   ^r'^  +  3  +  x^.  3.   xr^  ^jc'-V  19. 

4.    Prove  last  by  general  method  of  differentiation, 

28.  If  a  function  of  x  is  the  product  of  a  constant  by 
another  function  of  x^  i.e.  if 

F{x)  =  K<i>{x\  (i) 

then  F\x)  =  K<l>\x)',  (2) 

that  is,  the  derivative  of  the  product  of  a  constant  by  a  func- 
tion is  the  product  of  the  constant  by  the  derivative  of  the 
function. 

Proof.  —  When  x  becomes  x  -\-  Ax,  (i)  becomes 

F(x  +  Ax)  =  X(l>(x  +  Ax).  (i)' 

Subtracting  (i)  from  (i)'  and  dividing  by  Ax,  we  have 

F(x  +  Ax)  -  F{x)  _  K4>  (x  4-  Ax)  -  X<t>  (x) 

Ax  Ax 

j^<ti(x-{-Ax)  -<l>(x)  ^ 

=  A > 

Ax 

or  at  the  limit,  Fix)  =  K<f>'(x). 

Corollary.  —  The  derivative  of  mx""  is  m  times  the  de- 
rivative of  ^%  as  given  in  §  16.  Hence,  it  is  mnx^~^.  This 
result  is  so  often  used  that  it  should  be  carefully  memorized. 
When  n\^\,  the  derivative  is  simply  771.  (Show  this  directly, 
by  §  18.) 

Examples. 


•iffere 
5- 

:ntiate 

4■^^^ 

3^,  \^. 

3<^ 

3' 

y/2x^ 
5 

x4i 

+  - 

I 

V5 ' 
-V2, 

GENERAL    THEOREMS   OF  DIFFERENTIATION      23 

29.  If  a  function  of  x  is  the  product  of  two  functions  of 
X,  i,e.  if  Fix)  =  <j>(x){l/(x),  then 

F(x  +  Ax)  =  <f>(x-{-Ax)if/(x-{-  Ax). 

Subtracting  and  dividing  by  Ax,  we  have 

F(x  +  Ax)  —  F{x)  __<i>{x-\-  Ax)if/{x  +  Ax)  —  <l>  (x)  if/ (x) 

Ax  Ax 

The  right  member  may  be  changed  in  form  without  suf- 
fering any  change  in  value  by  adding  and  subtracting 
<^  (x)  i{/(x-\-  Ax)  in  its  numerator,  giving 

4>(x-\-Ax)^//(x-{-Ax)-<|>(x)^f^(x)-<|>(x)^|/(x-\-Ax)-]-(f>(x)^//(x+^x) 

Ax 

Grouping  the  terms  according  to  common  factors,  we 
have 

[<^  (x  +  Ax)-<I>  (x)']  if/  (x  -\-Ax)  +  <l>  (x)[i{/ (x  +  Ax)  -  if/jx)^ 

Ax  ^  ' 

or 

Ax  r\      "         /  ^^ 

Taking  these  terms  in  order,  we  see  that  the 

1-    v     .<l>(x-\-Ax)  —  <f>(x)  .       ,.  . 

hmit  of  -^ T^ — r^_z_  js  f^'(x), 

Ax  ^ 

limit  of  if/(x-\-Ax)         isi{/(x), 

,.    .      .  il/ (x -{- Ax)  —  d/ (x)    .      .,  , 

hmit  of  ^-^ — ■ — -^ ^-^^   is  i^\x)f 

Ax 

limit  of  <f>{x)  is  <^(^), 

which  gives  for  the  limit  of  the  right  member  of  the  equation 

<l>'(x)il,(x)-\-^{,\x)<f>(x); 


24  INFINITESIMAL    CALCULUS 

while  for  the  other  (or  left)  member  of  the  equation  the 

,.    .        F{x -\- t^x)  —  F{pc)  . 

hmit  of  -^ -^ ^^  IS  F'(x), 

Putting  these  limits  equal,  we  have 

F\x)  =  <l>Xx)  if;  (x)  +  ij;'(x)  <t>  (x). 

In  words,  ^/le  derivative  of  the  product  of  two  functions  is 
the  sum  of  the  products  obtained  by  multiplying  the  derivative 
of  each  function  by  the  other  function. 

dx  dx  dx 

=  2x(l-\-X^)+  2X'X^ 
=  2jr(l  +  2X^). 

Examples.  —  1.  Find  the  derivative  of  (i  -\-  x'^)(^i  ~  x'^)  first  by 
§  29  and  afterwards  by  multiplying  out  and  then  differentiating. 

2.  (2  +  ^5  -  ^)(5  +  ^),  4(^2  +  I.)  (^3  _  2), 

a{sx^  +  4)(Sx^+6x^  +  7x-\-S),  {a  +  d) {kx^^^  hx^-\- p) {qx"^  +  r). 

3.  Prove  §  28  by  using  §  29,  regarding  i  as  a  form  of  \^(;r)5  whose 
derivative  is  zero.     (See  §  27,  end.) 

4.  Prove  §  29,  using  a  different  notation. 

30.  Corollary. — If  F{x^-=fx(x)f^{x)fz{x')i  we  may  abbrevi- 
^\.^  fiipc)  fzix)  to  0(jr),  so  that 

whence  P{x)  =fi'{x)  4>{x)  +  <t>'(x)/i(x). 

Replacing  (f>(x)  by  its  value ^(;r)y^(.r)  and  <p'(^x)  by  its  value 

/2'(^)MX)+/S>(X)/,(X), 

we  Kave 

F'ix)  =/,'(x)  [/2  W/3  W]  +  [/2'(X)MX)  4-/3' W/2(^)]/i(^) 
=/l'W/2W/3W  +/2'W/3W/l(-^)  +/3'W/lW/2(*). 


gENERAL    THEOREMS   OF  DIFFERENTIATION    25 

By  successive  applications  of  §  29  this  theorem  can  be  generalized 
to  the  product  of  any  number  of  functions,  and  in  words  is  as  follows: 

The  derivative  of  the  product  of  any  number  of  functions  is  the 
sum  of. the  products  obtained  by  multiplying  the  derivative  of  each 
function  by  the  product  of  all  the  other  functions. 

Examples.  —  Find  the  derivatives  of 

(.^2  +i){x+  i){x  -  I),    .r3(^2  +  2x  +  3)(2;«:4  -  7)(4  -  ^). 

31  •    If  ■^{^')  —     ^    . ,  and  <i>{x)  is  not  zero,  then 


Fix  +  Ajc)  —  F{pc)  _  <^(x  +  Ajc)      <^{x) 
^x  Ax 

_  <l>(x)  —  <f>{x  +  Ax) 

~'  Ax  (fi{x)(j>{x-^Ax) 

_  ~^  <l}{x-{-Ax)  —  <f>(x) 

~  <j>{x)<l>(x-\-Ax)'  Ax 

or  at  the  limit    ^'W  =  -ft=-^2- <^'(^) 
l<t>{x)j 

That  is,  the  derivative  of  the  reciprocal  of  a  function  is 
minus  the  derivative  of  the  function  divided  by  the  square 
of  the  function. 


Thus  the  differential  quotient  of  — -  is 

3^ 


-^(3^2) 


^^         or  -^,  or  -  2 


(3;r2)2  9^*  3^ 

32.    Examples. 

1.    Find  the  derivative  of 


^'     1+^'     l+x  +  X^  x^  (I+^)''^  (l+X  +  *= 


26  INFINITESIMAL    CALCULUS 

2.  Show  by  method  of  §  29,  that  if 

'AW 

then  J..  ^t!±j=Lf±^ 

where  the  (;r)'s  are  omitted  for  brevity. 

3.  Prove  the  same  theorem  by  applying  results  of  §§  29,  31,  after 
throwing  —  in  the  form  0  — 

33*  We  may  interject  here  an  application  of  the  result  of  §  31 
to  generalizing  the  theorem  of  §  16.  The  differential  quotient  of  x^ 
was  there  obtained  only  under  the  restriction  that  «  be  a  positive 

integer.    But  if  «  be  a  negative  integer,  —  m,  then  x"^  becomes  — .    This 

x^ 
fraction  has  meaning  only  provided  the  denominator  is  nut  zero,  i.e.  x 
is  not  zero.     The  differential  quotient  becomes 
—  mx^-^ 

X'^rr^        ' 

which  reduces  to  —  mx-^-^  or  nx*^-^. 

That  is,  the  restriction  imposed  in  §  16  that  n  must  be 
positive,  may  be  removed. 
Examples. 
1.    Differentiate  x'^.  2.    Differentiate  3^"^ 

3.    Differentiate  — .  4.    Differentiate  -^. 

x-i  Sx^ 

34.  If  we  wish  to  differentiate  the  quotient  of  two  func- 
tions as  ^)  (,  we  can  do  this  by  combining  the  results  of 
§§  29  and  31,  for  the  quotient  may  be  written  <f>(x)  • 

Thus,  the   differential  quotient  of  "^    is  obtained  by  writing  it 

(l  +  x^) -'     Applying  the  theorem  for  products,  we  get 

I  —  x'^ 


d 
(l+^2)_ 


dx  '^{i-x^J         dx 


which  can  readily  be  reduced. 


GENERAL    THEOREMS   OF  DIFFERENTIATION    27 

If  the  student  prefers,  he  may  simply  memorize  the  result  of  example 
2,  §  32,  and  apply. 

35.  If  2  is  a  function  of  y,  and  y  of  ^,  an  increment  £^x 
of  X  produces  Ay  of  ^y,  which  in  turn  produces  A2  of  2. 

Evidently  ^  =  ^  .  ^. 

^x      A_y      A^ 

The  limits  of  these  magnitudes  (assuming  that  definite 
limits  exist)  will  therefore  have  the  same  relation,  viz. : 

d^_dz^     dy 
dx     dy     dx 
This  may  also  be  expressed  : 
If  F{x)  =  ^\_f{x)-\, 

then  F\x)  =  ^\f(x)\f{x). 

It  must  be  carefully  noted  that  0'[/(^)]  means  the  derivative  of 
0[/(jr)],  not  with  respect  to  x^  but  with  respect  \.o  f{x).     It  is  —  not 

dz       ...  ^0[/(^)]   ^  ^0[/(^)]  '^y 

-y,  or  agam  it  is       ,\,    /     not       '-^,       -*. 
dx  ^  df{x)  dx 

In  words,  the  derivative  with  respect  to  x  of  a  function  of 
a  function  of  x,  is  the  derivative  of  the  former  function  7m'tk 
respect  to  the  latter,  multiplied  by  the  derivative  of  the  latter 
with  respect  to  x. 

Thus,  if  jj/  =  (i  +  x'^Y^  ^-  maybe  found  by  denoting  (i  +  x^)  by  tv, 

and  then  finding  -^  from  y  =  w',  and  —  from  w=  I  -\-  x'^.     Whence 
t         7         7  dzv  dx 

dx      d-w      dx      ^  ov    -r      y 

But  the  use  of  w  is  quite  unnecessary,  and  the  student  should  learn 

to  dispense  with  it  as  well  as  with  y  also.     The  required  derivative  then 

Employing  the  notation  of  differentials,  the  process  is  even 
more  easily  remembered  and  applied.    The  differential  o< 

y^[/(^)]  or  ^'U{x-)yAx),  or  <A'[/(^)]/W«'.»^- 


28  INFINITESIMAL    CALCULUS 

That  is,  we  first  differentiate,  treating  "f{x) "  as  a  single 
character,  and  our  result  contains  dfi^x).  We  then  perform 
the  further  differentiation  indicated  by  this  df{x). 

Thus,  ^(  I  4.  y2) 3  ^  3  (  I  +  ^2)2^^  I  j^  ^2) 

=  3(1  -^x^Y^xdx, 

where  "  (i  +  x^y  is  first  kept  intact  as  if  it  were  not  a  combination  of 
symbols,  but  a  single  cumbrous  symboL 

36.    Examples. 

1.  Differentiate  4(2  +  x^^. 

2.  Differentiate  (7  + ^)^ 

3.  Differentiate  2(1  +  2^-!-  x^)^.      Ans.  12(1  -i-  ;r) ( i  -\-2x-\-  ^-)'. 

4.  Differentiate  (3  ;r^- 2)--*. 

5.  Differentiate . 

(^2  +  ^+1)2 

6.  Differentiate ^ .         Ans. 90£(-^_4il)__^ 

7.  Differentiate  a^b{\-\-  x'^y  +  ^(i  +  x"^)^  -\- k{i -\- x  f, 

8.  Differentiate 

37*    III  lil^6  manner,  if  we  have  a  function  of  a  function  of  a  func- 

we  may  show  that  F\x^=4>^{Xix')-\l\x\ 

Substituting  for  ^  its  given  value  and  for  ^'  its  value  as  obtained  by 
§  35>  we  have 

^w  =  0'{(V'[/w])}^'[y(-^)]/w, 

and  so  on  for  any  number  of  functions.     If  we  use  differentials  instead 
of  differential  quotients,  we  have 

40l(02[</>3(-)])}  =  <^l'^02 

=  0l'02'^03 

=  etc. 
The  proof  is  left  to  the  student. 


GENERAL    THEOREMS   OF  DIFFERENTIATION     29 

Examples. 

1.  Plnd  the  derivative  of 

4{2(I   +  ^2)-2  4-  3(1  +  ;,2)3|2  +  5(2(1   +  x'^Y  +  3(1  +  X^Yf. 

2.  Differentiate   {a+[d+{c  +  Ax")^Yf. 

38.   The  results  of  this  chapter  may  be  thus  summarized  : 


dx 


K4,'(x). 


'^'  dx     ~  l<t>{x)Y 


30  INFINITESIMAL    CALCULUS 


CHAPTER   III 

DIFFERENTIATION   OF   THE   ELEMENTARY   FUNCTIONS 

39.  We  have  learned  (§§  16,  33)  that  the  derivative  of 
x"^  is  «jc"~^,  where  n  is  any  integer,  x^  is  the  elementary 
algebraic  function. 

We  have  now  to  differentiate  elementary  functions  called 
"transcendental."  To  do  this  we  recur  to  the  general 
method  of  differentiation.  We  first  take  up  the  trigono- 
metric functions. 

^0»  ^(sin  x)  _  y     sin  (^x  +  A.r)  —  sin  x 

dx  Ax 

_y     sin  X  cos  Ax  -f  cos  x  sin  Ax  —  sin  x 


=  lim| 


Ax 

sin  A^        .  I  —  cos  Ax 

cos  X sm  X  • 

Ax  Ax 


But  becomes  unity  at  the  limit  when  Ax  becomes  zero,  and 

Ax 

-  cos  Ax  , 

becomes  zero. 


Ax 

These  are  shown  by  means  of  Fig.  4,  where  AB  is  an  arc  Ax  on  a 
unit  radius  OA.  So  that  ^C  is  sin  Ajf,  CO  is  cosAjt,  and  CA  is 
I  —  cos  Ax. 

!illA£  is  therefore  ^ 
Ax  BA 

and  i_:^cosA£  .^  CA^ 

Ax  BA 


DII^FERENTIATION  OF  FUNCTIONS 


31 


When  BA  becomes  zero,   CA  and  BC  become  zero.      The  proof 

,t  lim  — 
B. 
ing  hints : 


that  lim =  i,  and  lim =  o,  is  left  to  the  student  with  the  follow 

BA  BA 


i>  f  RC       CO 

1.    I  > > = ,  which  approaches  i  as  limit 

arc^^      DA      AO  ^^ 

2     CA^CAB£^B£B£    ^^^^^  approaches  o  x  i. 
BA      BC    BA      CE    BA  ^^ 


Hence 


</(sin  x) 
dx 


=  cos;r  X 


sm;rx  o 


=  cos  X. 
In  like  manner,  we  may  prove 
</cos  jr 


dx 


=  —  sm  X. 


41 


dtSLiix 


/sin^N 

\COS  X  / 


dx  dx 

cos  X  cos  jf  +  sin  jr  sin  x 


Similarly, 


</(cot  x')  _    -  1 
dx  sin^  X 


32  INFINITESIMAL    CALCULUS 


a  sec  X 


^  ^^      ^ 'd^ =  ^^^•'  according  to  §  31, 

^/(cosecjr)         Isin^ri 

and  -\ =  — ^— ; —  etc. 

dx  dx 


43.        ^     ^  =  hm 


dx  Ax 

Ax 


=  lim^'.^ 


Now  let  a^*  -  I  =  S,  so  that  a^*  =  i  +  5, 

and  Ax  log  a  =  log  (i  +  5), 

and  .  Ax  =  ^^g(^  +  ^X 


Then  ^  =  Uma^  ' 


log  a 

a^  — 
dx  log  ( I  +  5  ) 


log  a 
=  lim  a^  log  a 


log(i  +  5) 
5 


=  lim  a^  log  a j-. 

log  {(I  +  5)«} 
1 
The  limit  of  (i  +  5)^,  when  5  becomes  zero  (vvhich  evidently  occurs 
when  Ax  becomes  zero)  is  2.718  approximately,  and  is. called  <?.* 

*  This  fundamental  magnitude  may  be  pictured  as  follows:  Suppose 
interest  is  at  4%  corresponding  to  "  25  years  purchase."  $  i  com- 
pounded yearly  for  this  25  years  amounts  to  (1.04)25.  Compounded 
half-yearly  for  the  same  25  years,  it  is  (1.02) 5*^;   quarterly  (i.oi)i'^'^; 

daily  (i  +  s^f 00)  J  momently,  lim(i  +  5)  ,  or  e.  Thus  <?  is 
simply  the  amount  of  ^  i  at  ?nomently  or  <f<?«/m«<7?^j  interest  daring  the 
''purchase  period."  This  is  ^2.718,  whereas  with  quarterly  com- 
pounding the  amount  would  be  $  2.705,  and  with  yearly,  $  2.666. 


DIFFERENTIATION  OF  FUNCTIONS  33 


Hence  at  the  limits      ^^^^-^  =  «^  log  a 

dx  log  e 

This  result  is  independent  of  the  system  of  logarithms.  It  is  true 
of  "common  logarithms."  If  we  take  e  as  the  base  {i.e.  employ  the 
Naperian  system),  then  log  <?  =  i,  and  the  result  simplifies  to 

-^^ — ^  =  a*'  log  a. 
dx 

Finally,  \i  a  ■=.  e^  the  result  is  still  simpler,  for  log«^  =  i.     We  then 

dx 

Henceforth  we  shall  denote  common  logarithms  by 
"  Log  "  and  Naperian  logarithms  by  "  log."  Any  other  sort 
of  logarithms  will  be  denoted  by  "  logj,"  where  the  subscript 
b  denotes  the  base  of  the  system. 

44*  We  now  proceed  to  the  inverse  functions  of  those  just  considered. 
y  =  arc  sin  x,  means  that  y  is  the  arc  whose  sine  is  x  (sometimes 
the  notation  sin"i  x  is  used),  i.e.  it  means  the  same  thing  as 

X  =  sinjK- 

dx 
From  this  — -  =  cosjv 

dy 


VI  —  sin^jj/ 


—  Vi  —  X-. 

But  —   is  the  reciprocal   of  -^,  since  these   expressions   are   the 

dy  dx 

limiting  values  of     '^  and  -=^,  which  are  reciprocals. 
Ay  Ax 


Hence 
Or 

Similarly, 


^ 


Vi 

dx 
^(arc  sin  x)  _ 

I 

dx 
d{zxc  cos^f)  _ 

Vi 

-x^ 

dx  Vi  -  a^ 


34  INFINITESIMAL    CALCULUS 

45.    \iy  =  arc  tan  x,  then  x  =  tan^. 
i/x  I 


dy      co'^y 

=  sec^j 

=  I  +  izx^y 

=  i+;r2. 

Hence 

dy^       I      . 
^;«r       I  +  ;«:2 

Or 

</(arc  tan 
dx 

^)_       I 

l+;r2 

Similarly, 

</(arc  cot  jr)  _    —  I 
^;r              I  +x^ 

46.    If  ^  =  log^,  then  X  =  ^y,  where  b  is  the  base  of  the  system. 

Hence  ^=^logj^^^. 

dy  log6^ 

But  log6^  =  l. 

dy  logj/r 


Hence  ^  =  ^»     ' 


log6<r 
Hence  ^  =  128^. 

This  is  independent  of  the  particular  system  of  logarithms. 
If  ^  =  ^,  then  logft^  =  I ,  and  the  result  simplifies  to 

dy  \  J  dx 
-^  =  —  ovdy  =  — 
dx      X  X 

47  •    We  may  now  still  further  generalize  the  theorem  expressed  in 
§§  16,  33.    The  number  n  has  been  restricted  to  an  integer.     But  if 
y  =  x^  where  n  is  any  real  number^ 
then  log  y  =  n  log  x. 

Taking  the  differential  of  each  side, 
dy         dx 


DIFFERENTIA  TION  OF  FUNCTIONS  35 


Hence  ^  =  ^ 

dx      X 

X 


That  is,  the  restriction  of  §§  i6,  33,  that  n  must  be  an 
integer  is  now  removed.  It  may  be  a  fraction,  an  irrational 
number,  or  any  real  number  whatever. 

Examples. 

1.    What  is  the  differential  quotient  of 


r2,  x'^y  x^,   Vx,   Vx,  X  3^——? 
Vx 


2.    Of         Vi  +  x^,  (jfi  -  1)%  ^Ja-\-^^/x  +  cxh 

48.   The  results  of  this  chapter  may  be  thus  summarized 
Direct  Functions.  Inverse  Functions. 

d{mx^)  =  mnx**~^dx. 

^/(sin  x)  =  cos  xdx.  //(arc  sin  x)  = 


//(cos  x)=  —  sin  xdx.  //(arc  cos  x)  — 


—  dx 


Vi-^ 


//(tan  x)  =  — ~  //(arc  tan  x)  =  — ^• 

^        COS^^  1+^ 

^(cot  x)  =  =-^-  //(arc  cot  x)  =  ^=^' 
^         ^     sm^^  1+^'' 

^^^    ^^-Log^  ^(Log^)^^Log. 

Log^  X 

dx 
=  a'  log  adx.  //(log  ^)  =  — 


d(e^\  =  ffdx 


x 


36  INFINITESIMAL    CALCULUS 

No  function  inverse  to  x^  (or  to  the  more  general  form 

kx"^)  is  given,  since  in  this  case  the  inverse  is  identical  in 

form  with  the  direct  function. 
1 
(Thus,  \i y  =  ;r",  x  =i  y^  =  y"*,  a  form  identical  with  x**,  its  inverse.) 

49.    Examples. 

1.  Diflferentiate  3  sin  x. 

2.  Differentiate  i  —  asinx  +  d  cos  x. 

3.  Differentiate  2  sin  x  cos  x.     Ans.   2  cos  2  jr. 

4.  Differentiate  sin  x  tan  x. 

5.  Differentiate  cot  x  +  x^  cos  x. 

6.  Differentiate  log  x  +  tan  x  cos .«.     y4«j.   -  +  cos  Xo 

X 

7.  Differentiate  x'^a''. 

8.  Differentiate  {a  log  ^  —  ^;ir2  +  <:aa:)  (j  _  _^)^ 

9.  Differentiate  sin  3  x.     Ans.  3  cos  3  x. 

10.  Differentiate  cos  jr^. 

11.  Differentiate  tan  (i  +  .r  +  .^2).     Ans.  '  +  2  ^    _. 


COs2(l   ^  X  ^X^) 

13.    Differentiate  \o^x^  -\--  +  x  tan  (;i;  +  a^  -  arc  cos  3  jr). 


SUCCESSIVE  DIFFERENTIATION  3fi 


CHAPTER   IV 

SUCCESSIVE   DIFFERENTIATION MAXIMA   AND   MINIMA 

50.  The  derivative  of  2  x^  is,  as  we  know,  ^  x\  The 
derivative  of  8  x''  is,  in  turn,  24  x^.  The  derivative  of  a 
derivative  is  called  the  second  derivative  of  the  original 
function. 

When  F{x)  stands  for  the  original  function,  and  F\x^ 
for  its  derivative  (to  avoid  misunderstanding  we  must  now 
call  it  the  first  derivative),  then  F^\x^  denotes  the  second 
derivative,  and  F^'\x)  the  third  derivative  {i.e.  the  deriva- 
tive of  F"(x),  etc. 

Again,  if  we  use  the  notation  -^  for  the  first  derivative^ 

dx 

.   .    .    .       <l)    .    . 

the  second  derivative  is  evidently  — ^^ — -,  which  is  usually 

{iX  /  79  ■ 

abbreviated  to  -ih,  \  likewise  the  third  or     ^/  ^  is  written 
dx-  .  dx 

— ^,  and  so  on  to  -^,  —4,  etc. 
dxr  dx^    dx'' 


51  •    Examples. 

1.  What  is  the  third  derivative  of  jtr^? 

2.  What  are  the  2d,  3d,  4th  derivatives  o{  x^l 

3.  Differentiate  successively  ax^.  When,  if  ever,  will  the  answers 
become  zero?  What  sort  of  a  number  must  n  be  to  bring  about  such 
a  result  ? 

4.  Differentiate  successively  sin  x.  Ans.  cos  x,  —  sin  x,  —  cos  x,  sin  x 


38  INFINITESIMAL    CALCULUS 

6.  Dififerentiate  successively  tan  x. 

6.  Differentiate  successively  a"^- 

7.  Differentiate  successively  arc  sin  x. 

8.  Differentiate  successively  arc  tan  x. 

Ans         '  2.y  2(1  -  3 ^'^) 

9.  Differentiate  successively  log  x, 

52.  Just  as  the  first  derivative  threw  light  on  the  problems 
of  velocity,  tangential  slope,  etc.,  so  the  second  derivative 
will  illuminate  acceleration,  curvature,  etc. 

We  have  seen  that  if  for  a  falling  body  s=  16  t^,  then 


S-". 

(I) 

whence 

(^) 

We  may 

understand  this  result  better  if 

we 

designat 

e^. 

by  Vy  as  in  §  6,  so  that  (i)  becomes 

;z;  =  32/,  (i)' 

and  (2)  'ft^^^'  ^^^' 

dv 
where  —  is  evidently  simply 

dh 
for  both  are  mere  abbreviations  of 


dt 

What  does  equation  (2)  or  (2)'  mean?     —  means  the  rate 

dt 

at  which  the  body  is  gaining  speed.     It  is  clear  that  moving 


SUCCESSIVE  DIFFERENTIATION  39 

bodies  do  gain  or  lose  speed,  and  that  some  gain  or  lose 
faster  than  others. 

The  gain  or  loss  of  speed  has  nothing  to  do  with  how  fast 
a  body  is  going.  A  slowly  moving  body  may  be  gaining 
speed  very  fast,  while  a  fast  moving  body  may  not  be  gain- 
ing at  all,  or  may  even  be  losing  speed. 

If  we  use  the  term  velo  to  indicate  a  unit  of  velocity,  or 
one  foot  per  second,  we  know  from  (i)  that  a  body  which 
has  fallen  2  seconds  has  then  a  speed  of  64  velos,  while  at 
the  end  of  5  seconds  its  speed  is  160  velos.  Here  is  a  gain 
of  96  velos  in  3  seconds,  or  an  average  of  32  velos  per 
second. 

This  does  not,  of  course,  imply  that  the  body  had  gained 
at  the  rate  of  32  velos  per  second  all  the  time.  But  equa- 
tion (2)  tells  us  that  this  is  the  case.  A  falling  body  on  the 
earth  is  constantly  gaining  velocity  at  the  rate  of  32  velos 
per  second. 

Rate  oi  gain  of  velocity  is  called  acceleration,  and  we  see, 
therefore,  that  a  faUing  body  is  a  case  of  "  uniformly  accel- 
erated motion." 

Observe  that  the  acceleration  or  rate  of  gain  of  velocity  expressed  in 
32  velos  per  second,  cannot  be  expressed  as  any  number  of  feet  per 
second.  On  the  contrary,  substituting  for  the  word  "velos"  its  defi- 
nition "  feet-per-second,"  we  see  that  32  velos  per  second  is  32  feet  per 
second  per  second. 

If  the  distance  a  body  moves  in  time  /  is  not  16/2,  but  10/*,  then  its 
velocity  is  30  r'^,  and  acceleration  60/.  In  other  words,  its  acceleration 
in  this  case  depends  on  the  time.  If  the  body  has  fallen  2  seconds,  its 
acceleration  is  120  velos  per  second  ;  if  3,  180  velos  per  second ;   etc. 

53.  If  F{x)  expresses  the  ordinate  of  any  point  on  a 
curve  when  the  abscissa  is  ^,  we  have  seen  that  F\x) 
expresses  the  tangential  slope  at  that  point.  What  does 
F^\x)  represent?    Evidently  the  rate  at  which  that  slope  is 


40 


INFINITESIMAL    CALCUL  US 


changing  at  that  particular  point  as  x  increases.  It  denotes 
vvhat  we  may  call  the  curvature  at  that  point  with  respect  to 
the  axis  oi  x. 


Fig. 


A,  rate  of  gain  o*^  slope  positive  ;  B  ("  point  of  inflection  "),  zero 
C,  negative. 


Curvature,  however,  is  usually  measured  with  respect  to 
the  tangent  itself.  The  expression  for  this,  the  more  proper 
sense  of  curvature,  is  somewhat  more  complicated.  At  a 
point  when  the  curve  is  horizontal,  the  two  sorts  of  curva- 
ture are  identical. 

54.  When  the  curve  is  horizontal,  the  slope  of  the  tan- 
gent F'{x)  is,  as  has  been  seen,  zero.  But  the  curve  may 
be  horizontal  at  three  sorts  of  points  :  a  maximum  as  at  A 


Fig.  6.  — Points  of  zero  slope:  A,  maximum:  B,  horizontal  point  of  inflection; 
C,  minimum;   D,  maximum. 


and  D  (Fig.  6),  or  a  minimum  as  at  C,  or  a  horizontal  point 
of  inflection  as  at  B. 

A  maximum  point  on  a  curve  is  a  point  such  that  the 
ordinate,  or  y,  of  that  point  is  larger  than  the  ordinates  of 
points  in  its  neighborhood  on  either  side.     (The  phrase 


SUCCESSIVE  DIFFERENTIATION  41 

"  points  in  its  neighborhood  "  means  all  points  on  the  curve 
within  some  small  but  finite  distance  on  either  side.)  A 
minimum  point  is  one  whose  ordinate  is  less  than  the 
ordinates  in  its  neighborhood  on  either  side.  A  point  of 
inflection  is  one  where  the  neighboring  parts  of  the  curve 
on  opposite  sides  of  the  point  are  also  on  opposite  sides  of 
the  tangent  as  at  B  in  Figs.  5  and  6. 

In  the  neighborhood  at  the  left  of  a  maximum  the  slope 
of  the  curve  is  positive,  while  on  the  right  it  is  negative. 
For  a  minimum,  the  slope  is  negative  on  the  left  and  positive 
on  the  right.  For  a  horizontal  point  of  inflection,  the  slope 
is  positive  on  both  sides  or  else  negative  on  both  sides. 

It  is  to  be  observed  that  a  curve  may  have  more  than  one  maximum 
or  minimum,  and  that  a  maximum  ordinate  does  not  mean  the  greatest 
ordinate  of  all,  but  only  the  greatest  in  its  neighborhood.  Thus  the 
ordinate  at  Z^  is  a  maximum,  though  that  at  A  is  larger. 

55.  Dropping  the  symbolism  of  the  curve,  it  is  clear  that 
when  a  function  F{x)  reaches  a  maximum  or  minimum,  then 
F\x)—  o,  for  F\x)  represents  the  rate  of  increase  o{  F(x)y 
and  at  a  maximum  or  minimum  this  rate  is  zero. 

But  if,  conversely,  we  have  F\x)=o,  we  simply  know 
that  for  that  particular  value  of  x  which  satisfies  this  equa- 
tion F(x)  is  not  infieasing  nor  decreasing.  We  cannot  tell 
whether  it  is  a  maxi.num  or  a  minimum  or  an  "inflectional 
stationary"  value  {i.e.  one  such  that  F{x)  will  increase  for  a 
change  of  x  in  one  direction  and  decrease  for  a  change  oi  x 
in  the  other  direction). 

56.  Now  these  questions  can  be  settled  by  recourse  to 
the  second  derivative,  provided  this  is  not  also  zero. 

If  the  second  derivative  be  positive,  the  function   is   a 
if  it  be  negative,  it  is  a  maximum.     This  will  be 


42  INFINITESIMAL    CALCULUS 

clear  if  we  remind  ourselves  of  the  meaning  of  the  second 
derivative.  It  indicates  the  rate  of  change  of  the  slope.  If 
positive,  it  means  the  slope  is  increasing;  if  negative,  it 
means  the  slope  is  decreasing. 

If,  therefore,  at  a  point  where  the  first  derivative  or  slope 
is  zero,  the  second  derivative  or  "curvature"  (§53)  is  posi- 
tive, we  know  that  at  that  point  the  slope  is  increasing.  But 
as  its  present  value  is  zero,  it  must  be  changing  from  a  nega- 
tive to  a  positive  value.  This  can  evidently  only  occur  at  a 
minimum.  Per  contra,  if  the  second  derivative  is  negative, 
it  indicates  a  slope  growing  less,  i.e.  (as  the  slope  is  now 
zero)  changing  from  positive  to  negative.  This  evidently 
occurs  at  the  maximum,  and  nowhere  else. 

Thus,  take  the  function  ;r^  —  27^.  This  has  for  first  derivative 
2,x^  —  2'j,  and  for  second  derivative  6x.  Putting  the  first  expression 
equal  to  zero  and  solving,  we  find  j;  =  ±  3  ;  that  is,  the  function 
x^  —  2'j  X  has  two  points  at  which  it  is  stationary  (or  the  tangent  is 
horizontal),  where  x  is  3,  and  where  ;c  is  —  3.  The  first  of  these  is  a 
minimum,  and  the  second  a  maximum  ;  for  the  second  derivative  6  jt  is 
positive  for  ^  =  3,  and  negative  for  ;r  =  —  3. 

57.  The  exceptional  case  mentioned  in  §  56  (viz.  where 
the  value  of  x,  which  renders  the  first  derivative  zero,  also 
renders  the  second  derivative  zero)  seldom  occurs  in 
practice.  When  it  does  occur,  we  cannot  decide  the  nature 
of  the  function  for  that  point,  without  recourse  to  the  third 
derivative.  If  this  be  positive,  the  function  is  neither  at  a 
maximum  nor  minimum,  but  at  a  hori- 
zontal point  of  inflection,  as  at  A  (Fig. 
7),  when,  for  an  increase  of  x,  the 
Fig.  7.  function   was   increasing,   both   before 

and  after  the  point.  If,  on  the  other  hand,  it  be  negative, 
the  function  is  at  a  horizontal  point  of  inflection  as  at  B 


SUCCESSIVE  DIFFERENTIATION  43 

(Fig.  6),  when  the  function  was  decreasing  both  before  and 
after  reaching  this  point.  If,  finally,  it  be  zero,  we  are  again 
left  in  the  dark  as  to  the  nature  of  the  function,  and  must 
proceed  to  the  fourth  derivative.  We  employ  this  just  as  if 
it  were  the  second.  If  it  turns  out  zero,  and  forces  us  to 
consider  the  fifth,  we  employ  this  just  as  if  it  were  the  third, 
and  so  on. 

That  is,  as  long  as  the  successive  derivatives  turn  out  zero, 
we  go  on  until  we  find  one  which  is  not  zero.  If  this  deriva- 
tive be  of  an  even  order  {i.e.  2d,  4th,  6th,  etc.,  derivative), 
we  know  that  the  function  is  either  a  maximum  or  a  mini- 
mum^ and  is  the  one  or  the  other  according  as  the  derivative 
in  question  is  negative  or  positive.  But  if  the  derivative 
which  does  not  vanish  is  of  an  odd  order  {i.e.  3d,  5th,  etc.), 
we  know  that  the  function  is  neither  at  a  maximum  or  mini- 
mum value,  but  at  a  point  of  horizontal  inflection  and  is 
increasing  or  decreasing  according  as  the  derivative  is  posi- 
tive or  negative. 

58.  We  shall  not  devote  the  requisite  space  here  to  proving  the 
truth  of  the  last  section  in  full,  but  shall  merely  indicate  the  first  step, 
leaving  the  student,  if  he  so  desires,  to  extend  the  demonstration. 

Suppose  in  testing  the  function  F\x^  we  find  for  the  value  of  x 
which  renders  F\x^-=Oy  that  /"(^r)  is  also  zero,  but  F'"{x)  is  posi- 
tive. Denoting  this  value  of  x  by  xi,  we  may  state  the  problem  as 
follows  :  given 

F'{xi)    =0, 
F"(xi)  =  o, 
E"'{xi)>o, 
to  discover  the  nature  of        F(xi). 

We  shall  solve  this  by  reasoning  from  F'"  successively  back  to  F"f 
F',  and  F. 

Since  F"'(xi)  is  positive,  it  shows  that  F"(x)  is  increasing  as  x 
increases.     But  as  F"(xi)  is  zero,  the  fact  that  F"(x)  is  increasing 


44  INFINITESIMAL    CALCULUS 

shows  that  it  was  negative  before  reaching  F"{xi)  and  positive  after. 
This  is  our  conclusion  for  F". 

Since  F"{x)  was.  negative  before  reaching  F"{xi)  it  shows  that 
F'(^x)  was  //len  decreasing,  and  since  F"{x)  was  positive  afterward, 
F'{x)  was  /Aen  increasing. 

But,  if  F'(x)  is  zero  at  F'(x{)  and  was  decreasing  before  and  in- 
creasing after,  it  must  have  been  positive  both  before  and  after.  This 
is  our  conclusion  for  F'.  Since  F'  is  positive  both  before  and  after, 
it  shows  that  F(^x)  was  increasing  both  before  and  after,  and  is  there- 
fore not  a  maximum,  but  a  horizontal  point  of  inflection. 

Thus  let  F{x)  be 

x*-6x^  +  Sx+  7. 

Then  F'  is  ^x^  —  I2.r  +  8. 

Then  F"  is  I2;»:2-  12. 

Then  F'"  is  24  jr. 

The  roots  of  F'  =  o  are  i  and  —  2.  For  jr  =  i,  F"  vanishes,  but 
F'"  is  positive.  Hence  we  know  that  /'  or  x*  —  6^^^  -f  8;f  -f  7  is  at  a 
stationary  inflectional  value  increasing  on  either  side,  as  x  increases. 

But  for  ^  =  —  2,  F"  is  positive.  Hence  for  this  value  of  x,  F  is  a. 
minimum. 

59*    Examples. — 1,    Find  maximum  or  minimum  value  of  .j:^. 

2.  Find  maximum  or  minimum  value  of  ^x^  —  27  x. 

3.  Find  maximum  or  minimum  value  of  2x'^  -{-  x  -\-  i. 

4.  Find  maximum  or  minimum  value  of  .r^  —  12  jt  +  6. 

5.  Find  maximum  or  minimum  value  of  2 x^  ■}-  6 x'^  -{-  6 x  +  ^. 

6.  Find  maximum  or  minimum  value  oi  x^  —  2x  -{-  ^x^  —  4. 

7.  What  is  the  nature  o{  x*  —  24.  x^  -{-  64  x  -{■  10  for  x  —  2? 

8.  What  is  the  nature  oix^ -{-4x^  +  6  x^-{-4x+  ij  ior  x  =—  i  ? 

60.  If  F(x)  is  of  the  form  (f>(x)  -f-  X,  where  JC  is  any 
constant,  then  the  same  values  of  x  render  J^(x)  a  maxi- 
mum or  minimum  as  render  cf)(x)  a  maximum  or  minimum 
respectively. 


SUCCESSIVE  DIFFERENTIATION  45 

For  the  nature  of  F{x)  or  of  <t){x)  as  to  maxima  and  minima  de- 
pends exclusively  on  the  nature  of  their  derivatives,  and  the  derivatives 
of  these  two  functions  (viz.,  0(;c)  +  iTand  0(^))  are  evidently  identical 

Thus  to  find  the  value  of  x  to  render 

a  maximum  or  minimum,  we  may  drop  the  constant  term  and  simply 
inquire  for  what  value  of  x  the  form  x^  is  a  maximum  or  minimum. 

6i.  li  F{x)  is  of  the  form  K<j>{x)  when  X  is  a  positive 
constant,  then  the  values  of  x  which  render  F{x)  a  maxi- 
mum or  minimum  are  the  same  as  those  which  render  <^{x) 
a  maximum  or  minimum  respectively. 

\{  F{x)  =  K<^{x)  where  ^  is  a  negative  constant,  then 
the  values  of  x  which  render  F(x)  a  maximum  or  minimum 
are  the  same  as  those  which  render  <^(^)  a  minimum  or 
maximum  respectively. 

For  the  successive  derivatives  of  these  two  functions  (viz.,  K<t>{x) 
and  (t>{x))  are 

K<p\x)    )  [<p'{x\ 

K<t>"{x)       and      <t>"{x), 
etc.       J  [     etc., 

and  evidently  the  very  same  values  of  ;*:  will  make  the  two  first  deriva- 
tives zero,  and,  if  K  be  positive,  will  make  the  two  second  derivritives 
of  the  same  sign  or  both  zero;  but  if  K  be  negative,  will  make  them  of 
the  opposite  sign  or  both  zero.  Similarly  for  the  two  third  derivatives, 
etc.  Since  the  natures  of  /^and  of  0,  as  respects  maxima  and  minima, 
depend  exclusively  on  the  signs  (-f,  — ,  or  o)  of  their  derivatives, 
the  theorem  is  proved. 

Thus,  to  obtain  the  value  of  x  which  will  make 

a  maximum  or  minimum,  we  drop  the  constant  factor  (which  is  evi- 
dently positive)  and  find  out  which  values  of  x  make  x'^  —  x,  a.  maxi 
mum  or  minimum. 


46  INFINITESIMAL   CALCULUS 

Examples.  —  1.    Interpret  the  theorems  of  §§  60,  61  geometrically. 

2.  Find  maximum  or  minimum  of  5(1  -\-  x  ■\-  x^^-\-  10. 

3.  Find  maximum  or  minimum  of  —  3;i:(;r+i+  —  J. 

4.  Find  maximum  or  minimum  of  m  |  ^(-^   ^bx-\-  c')-^  e  _,_  ^ | . 


62.  The  subject  of  maxima  and  minima  is  one  of  the 
most  important  in  the  Calculus,  and  has  innumerable  appli- 
cations in  Geometry,  Physics,  and  Economics. 


Let  ABC  (Fig.  8)  be  any  triangle,  and  EFKH  a  rectangle  in- 
scribed within  it.  This  inscribed  rectangle  will  vary  in  size  according 
to  its  position.  If  too  low  and  flat,  it  is  small.  If  too  high  and  thin, 
it  is  also  small.  Between  these  positions  there  must  be  a  position  of 
maximum,  where  the  area  is  the  largest  possible. 

Now  its  area  is  the  product  of  the  base  HK  or  EF  by  the  altitude 
DM,  and  the  problem  consists  in  discovering  where  EF-  DM  is  a 
maximum. 

To  do  this,  we  must  first  express  EF  and  DM  in  terms  of  some  one 
variable.  Out  of  the  many  possible  {e,g.  BH,  BK,  AE,  EC,  EH,  HK, 
etc.)  we  select  AM,  and  denote  it  by  x.  We  call  AD  =  ^  and  BC=:a. 
Evidently  MD  =  h  —  x.  To  express  EF  in  terms  of  x,  we  proceed  as 
follows  :  The  triangles  AEF  and  ABC  are  similar,  so  that  their  bases 
and  altitudes  are  proportional.     That  is, 

AM^EF  ^^  x^EF 
AD      BC   ""^  h        a  ' 


SUCCESSIVE  DIFFERENTIATION  47 

whence  EF=  — • 

h 

Consequently  EF  X  DM  -{h-x)  — • 

h 

We  wish  to  know  for  what  value  of  x  this  expression  is  a  maximum. 

We  may  omit  the  positive  constant  factor  -,  leaving 

h 

{h  —  x)x  or  hx  —  x^y 

the  first  differential  of  which  is    h  —  2.Xy 

which,  put  equal  to  zero  and  solved,  gives 

k 
x  =  -t 

2 

the  required  answer. 

We  are  sure  it  is  a  maximum  and  not  a  minimum  or  stationary  in- 
flectional value,  since  the  second  differential  is  —  2;   i.e.  negative. 

We  have  learned,  therefore,  that  the  maximum  rectangle  inscribed 
in  a  triangle  is  that  whose  altitude  is  half  the  altitude  of  the  triangle. 

In  physics  many  important  principles  depend  upon  max- 
ima and  minima.  Thus  the  equilibrium  of  a  pool  of  water,  a 
pendulum,  a  rocking  chair,  or  a  suspension  bridge,  is  deter- 
mined by  the  condition  that  the  centre  of  gravity  in  each 
case  shall  be  at  the  lowest  possible  point. 

In  economics  we  have  the  principle  of  maximum  con- 
sumer's rent,  of  maximum  profit  under  a  monopoly,  etc. 

63.    Examples. 

1.  How  must  a  given  straight  line  be  divided  so  that  the  product 
of  its  two  parts  shall  be  a  maximum  ? 

2.  What  is  the  minimum  amount  of  tin  necessary  to  make  a  cylin- 
drical vessel  which  will  have  a  given  capacity  A?  What  must  be  the 
relation  between  the  height  k  and  the  radius  of  the  base  r? 

3.  Find  the  maximum  cylinder  inscribed  in  a  circular  cone  of 
revolution.     Ans.  Altitude  of  cylinder  equals  one  third  that  of  the 


48  INFINITESIMAL    CALCULUS 

4.  Find  the  maximum  rectangle  inscribed  in  a  semicircle. 

Ans.  The  sides  are  -  \/2,  and  r^J~2, 

2 

5.  A  cylinder  of  revolution  has  a  given  diameter.  What  altitude 
must  it  have  in  order  that  it  may  have  the  least  total  area  in  propor- 
tion to  its  volume? 

Hint.  —  Express  volume  and  total  area  in  terms  of  the  variable  alti- 
tude Xy  and  the  constant  radius  r.     Then  find  v\  hen 

total  area  .  .   . 

IS  a  minimum. 

volume 

6.  If  the  function  pF{p)  is  continuous,  what  equation  gives  a  value 
of/  which  makes  the  function  a  maximum? 

Write  the  algebraic  expression  denoting  the  condition  under  which 
the  value  of/,  in  the  equation  asked  for,  corresponds  to  a  maximum  or 
minimum. 

7.  If  the  price,  /,  of  an  article  is  fixed  and  the  cost  of  producing  it, 
for  a  given  individual,  is  a  function  IX-^),  of  the  quantity  produced, 
X,  how  much  must  he  produce  to  make  his  profit,  xp  —  F{x),  a  maxi- 
mum or  minimum?  Express  this  result  in  words.  What  condition 
must  L\x)  satisfy  that  the  profit  may  be  a  maximum  and  not  a  mini- 
mum?    Express  this  condition  in  words. 

8.  Four  equal  squares  with  side  x  are  removed  from  the  corners 
of  a  square  piece  of  cardboard  with  side  c  and  the  sides  are  turned  up 
so  as  to  form  an  open  square  box.     If  the  square  box  is  to  be  of  maxi- 

£ 

6* 

9.  The  distance  between  two  points,  B  and  C,  on  a  coast  is  5  miles. 
A  person  in  a  boat  is  3  miles  distant  from  B,  his  nearest  shore  point. 
Supposing  he  can  walk  5  miles  an  hour  and  can  row  4  miles  an  hour, 
what  distance  from  C  should  he  land  in  order  to  reach  C  in  th**, 
shortest  possible  time  ?     Ans.  i  mile. 

10.  Given  /,  the  slant  height  of  a  right  cone;  find  the  altitude  when 
the  volume  is  a  maximum.     Ans.   -  y/^. 


TAYLOR'S  THEOREM  49 

CHAPTER  V 

Taylor's  theorem 

64.  We  know  that  certain  functions  can  be  developed  in 
terms  of  powers  of  variables.  Thus  {a-\-xy  becomes  by 
the  binomial  theorem 

a*-\-4  a^x  -f-  6  c^x-  +  4  ao^  +  x^. 

Again,  by  simple  division,  we  may  show  that  (provided  x 
lies  between  —  i  and  +  1) 

=  I  —  X  -\-  x^  —  x?  -\ . 

I  -\- X 

Now  the  Calculus  supplies  a  much  simpler  and  more  gen- 
eral method  than  algebra  of  developing  functions  in  series 
of  this  sort. 

Thus,  let  <^(^)  be  any  function  of  x  developable  in  the 
form 

<t>(x)=  A  4-  B{x  -  «)  +  C{x  -  af  +  D{x  -  of  +  .••, 
where  a.  A,  B,  C,  etc.,  are  constants,  and  the  series  con- 
verges.    We  shall  show  how  to  express  the  "  undetermined 
coefficients  "  A,  B^  C,  etc.,  in  terms  of  the  single  constant  a. 
By  successive  differentiation,  we  have  * 

<^\x)  =  B  ^  2  C{x  -  a)^  zD{x  -  of  -\-  '" 
<f>"{x)=      +2C  -f-2  .  3Z>(^-d;)^-... 

etc. 

*  By  §  26  which  can  readily  be  extended  so  as  to  apply  to  an  infinite 
number  of  terms  if,  as  is  here  assumed,  the  sum  of  these  terms  con- 
verges. 


50  INFINITESIMAL    CALCULUS 

Since  these  equations  (and  the  original  from  which  they 
are  derived)  are  true  for  any  value  of  x,  they  are  true  when 

They  then  become 

<f>(a)=J,  or     A=<f>(a)', 

<l>'(a)=i.B,  B=ct>'(a); 

etc., 
where  2  !  means  i  •  2  and  3  !  means  i  •  2  •  3,  etc. 
Substituting  these  values  of  A,  B,  C,  D,  etc.,  we  have 

«^  (^)  =  </>  (^)  4- <^' («)  (^  -  ^)  +  <^"  («)  i^^=;^' 

2  ! 

65.  This  series,  which  is  "Taylor's  theorem,"  expresses 
the  magnitude  of  the  function  </>  for  any  value  of  x  in  terms 
of  its  magnitude  and  that  of  its  derivatives  for  any  other 
value  of  X. 

Thus  if  we  could  write  down  some  exact  formula  y  =  (t>  (x)  for  the 
population  {y)  of  the  United  States  in  reference  to  the  time  (jr) 
elapsed  since,  say  1800,  Taylor's  Theorem  tells  us  that  we  could  get 
the  population  in  1900,  0  (;f),  merely  from  data  of  the  census  of  1890. 

As  a  first  approximation  we  take  the  population  of  1890  itself,  0  (a). 
But,  as  the  population  has.  not  remained  stationary,  we  add  a  correction 
for  the  increase  within  the  decade. 

This  increase  we  first  assume  to  be  (^x  —  a)  <p'(a),  i.e.  the  rate  of 
increase  known  to  exist  in  1890,  <j>'{a)y  multiplied  by  the  time  between 
the  two  censuses  {x  —  a).     But  since  the  rate  of  increase  (by  which  is 


TAYLOR'S   THEOREM 


51 


here  meant  so  many  thousand  souls  per  year,  not  the  percentage  rate) 
has  not  remained  stationary,  we  add  another  correction       ^^'^^  ~  ^J  , 

I  •  2 

constructed  on  the  supposition  that  the  rate  of  increase  of  the  rate  of 
increase  of  population,  <f>"(a),  known  to  exist  in  1890  has  remained 
constant  until  1900.  Not  content  with  this,  we  take  into  account  the 
rate  of  increase  of  the  rate  of  increase  of  the  rate  of  increase  of  popu- 
lation, and  so  on. 

66.  Geometrically,  the  theorem  states  that  the  ordinate 
of  any  point  of  the  curve  y=  <l>{x)  can  be  obtained  from 
the  ordinate,  slope,  "  curvature,"  etc.,  of  any  other  point. 


Thus,  OB  (Fig.  9)  is  x  and  BD,  <f>(x);  OA  is  a  and  AC,  <p(a). 
The  theorem  tells  us  that  the  ordinate  of  the  point  Z>  can  be  ascer- 
tained purely  from  the  data  as  to  the  curve  at  C,  viz.  its  height,  the  rate 
at  which  this  height  is  increasing  (i.e.  its  slope),  the  rate  at  which  this 
slope  is  increasing  (i.e.  its  "curvature"  (§  53)),  the  rate  at  which 
this  "  curvature  "  is  increasing,  etc.,  etc.  In  fact,  the  theorem  states 
that  the  ordinate  DB  is  the  sum  of  various  magnitudes:  first,  0(rt), 
which  is  represented  by  ^5  (for  this  is  the  same  as  AC);  secondly, 

58' 
(x  —  a)<f)'(a),  which  is  represented  by  55'  (for  —  is  the  slope  of  the 

C5 


52  INFINITESIMAL    CALCULUS 

curve  at  C,  and  so  =  0'(a),  hence  55'  =  C5  X  <t>\a)  =  (x  —  «)  (l)'{a)), 

thirdly,  ^^ j  ^   k    j ^  which  is  represented  by  5'5",  when  5"  is  reached 

2 ! 
by  drawing  the  curve  C5",  which  has  the  same  curvature  as  the  prin- 
cipal curve  CD  has  at  the  point  C,  but  retains  that  "curvature"  (with 
respect  to  the  jr-axis,  see  §  53)  throughout;  that  is,  we  approach  D  by 
adding  successive  corrections.  5  is  the  position  Z>  would  have  had  if 
the  ordinate  of  the  curve  had  remained  unchanged  from  C  (so  that  the 
curve  would  have  followed  the  horizontal  C5) ;  5'  is  the  position  D 
would  have  had  if  the  rate  of  increase  of  the  ordinate,  i.e.  the  slope 
of  the  curve,  had  remained  unchanged  from  C  (so  that  the  curve  would 
have  followed  C5') ;  5"  is  the  position  D  would  have  taken  if  the  rate 
of  increase  of  the  slope  had  remained  unchanged  from  C  (so  that  the 
curve  would  have  followed  C8"),  etc. 

67.  If  we  take  the  point  li  instead  of  C,  so  that  a  =  o, 
Taylor's  theorem  reduces  to  the  simple  form 

<j>{x)  =  <i>{p)  +  <i>\o)x  +  ^  \        +^    \^      +etc. 

2  !  3  ! 

This  is  Maclaurin's  Theorem. 

The  student  vi'ill  observe  that  </>(o)  is  by  no  means  itself  zero.  It  is 
simply  that  particular  value  of  <p{x)  obtained  by  putting  x  =^  o.  Thus, 
if  0  {x)  is  x^  +  2x'^  +  117,  0(0)  is  117. 

68.  A  second  mode  of  stating  Taylor's  Theorem,  and  one 
often  met  with,  is  obtained  by  denoting  the  difference  of 
abscissas  x  —  a  hy  h,  and  replacing  x  by  a  -\-  h  (for,  if 
X  —  a  =  h,  X  =  a  -{-  h),  so  that 

or,  changing  our  notation  from  a  to  x, 

<f>(x  +  A)  =  cf>(x)  +  <t>\x)k  +  <^"(:r)^+  -, 

2  ! 

where  x  now  refers  to  the  abscissa  of  C  instead  of  that  of  Z>. 


TAYLOR'S   THEOREM  53 

The  student  will  also  sometimes  see  the  theorem  expressed 
in  the  same  form,  but  with  y  employed  in  place  of  h. 

69.  There  are  *many  applications  of  Taylor's  theorem  in 
economics.  Cournot  in  his  Principes  Mathhnatiques  makes 
frequent  use  of  it,  as  does  Pareto  in  his  Cours  d'economie 
politique. 

When  /^  is  a  small  quantity,  as  in  some  of  Cournot's  cases 
of  taxation,  then  the  higher  powers  of  h  may  be  neglected, 
and  we  have  the  approximate  formula 

i^i^x  ■\-h)  =  <f>(x)-\-h<t>'(x). 

This  is  assuming  that  if  the  interval  AB  is  very  small,  the 
point  8'  will  coincide  approximately  with  D. 

70.  It  will  be  observed  that  an  hiatus  was  indicated 
in  the  demonstration  of  Taylor's  Theorem.  This  means 
that  it  is  not  always  possible  to  develop  <t>{x)  in  the  series 
proposed,  and  that  the  attempt  to  do  so  will  give  a  diverg- 
ing or  indeterminate  series. 

It  is  impossible  in  so  elementary  a  treatise  as  this  to  indi- 
cate in  what  cases  Taylor's  Theorem  is  applicable.  The 
subject  is  one  of  great  difficulty,  and  some  of  the  most  im- 
portant conclusions  relating  to  it  have  only  recently  been 
discovered. 

71.  To  show  the  application  of  Taylor's  and  Maclaurin's 
theorems,  let  us  use  them  to  develop  the  function  (a  +  xy, 
assuming  it  developable.     Since  <l>{x)  —  (ai-  x^, 

<t>\x)  =  n{a-\-xy-\ 

<f,"(x)  =  n(n-  i)(a-{-xy-\ 

etc. 


54 

INFINITESIMAL   CALCULUS 

Hence 

<^(o)=^", 

<^\6)=na^-\ 

<l>%o)=n(n-i)a^-\^ 

etc. 

Hence 

A(x^ -  <h(6) 4-  <h'(o)x  +  't>"(oy  , 

2   ! 

a  result  which  we  already  know  by  the  binomial  theorem. 

Again  let  us  develop  sin  x,  assuming  it  developable. 

Since  4>  (x)  =  sin  ;r  (p  (o)  =  o, 

<f>'(x)  =  cos;r  0'(o)  =  I, 

0''(;r)  =  — sin;r  <f>"(o)  =  o, 

<l>i"(^x)  =  -  cos  ;r  0'"(o)  =  -  i. 

etc.  etc. 

Hence 

x^ 
=  o  +  x  +  o-  —  -{-'- 

3' 
x^   ,  x^      x"^   . 

Again  let  us  take 


X  —  a  -\-  1 

Since  0  (;»:)  = ,  0(a)  =  I, 

;r  —  a  +  I 

0'(;c)  =  -  (;r  -  a  +  i)-2,  0'(a)  =  -  I, 

^"(x)  =2{x-  a+  i)-\  <f>f'(a)  =  2, 

0^"(^)  =  -2  .  3(^  -  a  +  i)-4,  0''''(a)  =  -  3 !. 
Hence,  by  Taylor's  Theorem, 

<t>(x)  =  I  —{x  —  a)  +  -^ ^ ^-^^ ^  +  •••• 

2!  3! 


TAYLOR'S   THEOREM  55 

72.    Among  other  important  uses  of  Taylor's  and  Maclaurin's  theo- 
rems are  the  evaluations  of  the  fundamental  constants  e  and  t. 
To  obtain  ^,  we  develop  the  function  ^. 

0(^)=^,  0(0)=  I, 

0'(^)=^.  0'(o)=i, 

0"(^)=^,  0"(o)=i, 

etc.  etc. 

Since      ^(.)=,(o)  +  *'(o).  +  *:^  +  «f!+.., 

we  have  <r*=i  +;rH 1 — -  ■\-  •••. 

2      3! 

If,  in  this  equation,  we  put  x  =  i,  we  have 

2     3!     4! 

from  which  e  may  be  computed  with  any  required  degree  of  approxi- 
mation.    ^  =  2.71828  •••. 

To  obtain  tt,  develop  arc  tan  x. 

<p{x)  =  arc  tan  Xy  0(o)  =  0» 

0'(^)  =  — L_  0'(o)=i. 

I  +x^ 
If  ;r  be  less  than  unity,  we  know  by  algebra  that  * 

*^'w=rT^= '  ~  "'^  "^  ""^ " -^  +  •••• 

Hence    <t>"(x)  = -2x  +  4x^  -  6x^  + -",  0"(o)  =  o, 

iP'"{x)=-  2  +  3  .4^2  _  5  .6^4  +  ...,      0'"(O)  =  -  2, 

0<t>(;^)  =2.3.4^^-4.5.6^+    ....,  ^^''(O)  =   O, 

<f>f>{x)  =  2  .  3  . 4  -  3 . 4  •  5  •  6^'  +  •  ••»        ^"(o)  =  +  4 '.. 
etc.  etc. 

—  2X^  4'  X^ 

arctan;r  =  o  +  x  +  o -\ ; ho  +  ^^^^H 

3!  5! 

=  ,_f^  +  ^_£!+.... 
3       5       7 
*  It  is  assumed  here,  without  proof,  that  the  proper  conditions  as  t<j 
convergence  are  fulfilled. 


56  INFINITESIMAL    CALCULUS 

Let  X  be  — ^,  so  that  arc  tan^,  the  arc  whose  tangent  is  -^,  is  ^ 

V3  V3        6 

(2.(?.  an  arc  of  thirty  degrees).     The  preceding  equation  then  becomes : 

7r_^ I  I 


V3L       3-3     5-3'     7-3' 
3-3    '5  •3'     7-3 


whence  tp  ==  2V3     i ? 1 ? ? f-  ...  1 

L         3-3     'S-3''      7-3^  J 


=  3.14159.... 

73'    Examples. 

1.  Develop  («  —  x)'"^  in  series  of  ascending  powers  ot  x. 

2.  Develop  Va  —  ;f. 

3.  Develop  cos^f.     Ans.   i  -  •^-' +  •?!  _  "^^  .... 

^  2!      4!      6! 

4.  Develop  log  (i  +-*■)• 

6.   Dev'eiop  a^-^*. 

6.  Develop  ^.     ^;/j.  i  f  3  ^  +  ^^-  +  ^^• 

7.  Develop  K'^^  -  ^~*)- 

8.  Develop  arc  sin  ;ir.     Ans.x-\-~'~    ^ — -A ^-^  . U  .. 

2     3       2.4     5       2.4.6     7  ^ 

9.  Develop  cos  %. 

10.  Develop  ^  sec  j;. 

•  ./K^       iir^       )t^ 

11.  Develop  log  (i  +  sinx).     Ans.  x  — V  — 1-  ...„ 

12.  Develop  arc  tark^. 

13.  Develop  cos  {x  ■\-  y). 

Ans.  cos X  —  y  sinx  —  ^ cos ;r  +  —  sin  r  4-  •• 
2!  3! 

14.  Develop  tan  (x  -\-  y). 

1/3 

Ans,  tan  j^  +  >'  sec^jr  +  y^  sec^jt  tan  jr  +  —  sec2^(l  +  3  tan^^)  +  »- 

o 


INTEGRAL    CALCULUS  57 


CHAPTER   VI 

INTEGRAL   CALCULUS 

74.  We  have  thus  far  been  occupied  with  the  derivation 
from  F  of  F\  F'\  etc.  But  it  is  possible  to  reverse  this 
process,  and,  given  F'^\  or  any  other  derivative,  to  pass  back 
to  F^\  F\  F. 

F\x)  was  called  the  derivative  of  F{x)  ;  we  now  name 
F(x)  the  primitive  oi  F'(x).  The  first  process  of  obtaining 
F'  from  F  is  the  subject  matter  of  the  differential  calculus, 
of  which  the  preceding  chapters  have  treated.  The  process 
of  obtaining  F  from  F^  is  the  subject  matter  of  the  integral 
calculus, 

75.  In  the  differential  calculus,  we  saw  that  the  result  of 
differentiation  was  expressed  either  in  the  differential  quo- 
tient F'(x),  or  in  the  differential  F\x)dx.  In  the  integral 
calculus  it  is  customary  to  employ  only  the  latter  form.  We 
called  F\x)dx  the  differential  of  Fix)  ;  we  now  call  Fix) 
the  integral  of  F\x)dx.  We  obtained  F'(x)dx  from  F{x) 
by  differentiation.  We  obtain  Fix)  from  F\x)dx  by  inte- 
gration. The  symbol  of  differentiation  was  d ;  that  of  in- 
tegration is  j  . 

Knowing  that  dixr)  =  2xdx^  we  may  write  j  2xdx  =  :^  \ 
or  again,  since 

dFix)=.F\x)dx 


58  INFINITESIMAL    CALCULUS 

expresses  in  the  most  general  manner  the  process  of  the 
differential  calculus, 


^F\x)dx  =  F{x) 


expresses  the  process  of  the  integral  calculus.  Both  equa- 
tions state  the  same  fact  looked  at  from  opposite  directions. 
The  former  equation  reads,  "  the  differential  of  F{x)  is 
F\x)dx^^;  the  latter  may  be  read,  "the  function-of-which- 
the-differential-is  F\x)dx  is  F{x)j'  for  the  hyphened  words 
are  what  is  meant  by  "  integral  of." 

The  simplest  form  of  the  above  equation  is  \  dx  =  x, 

»75.    The   symbol    i    was  originally  a  long  S,  which  was  the  old 

symbol  for  "  sum  of"  (to-day  it  is  usual  to  employ  the  Greek  S  instead). 
Integration  was  looked  upon  as  summation,  dy  being  the  limit  of 
Ay,  and  Ay  being  a  small  part  of  y,  the  differential  dy  was  conceived  of 
as  an  infinitesimal  part  of  y.  An  infinite  number  of  dy's  were  thought 
of  as  making  up  the  y. 

77.   As  d{x^)  =  ^x^dx,  it  follows  that 

I  ^x^  dx  =  :x^. 

But  d(x^  +  5)  =  3  ^Vjc  ; 

hence  I  ^x^  dx  =  x^  -{-  $  ; 

that  is,  the  integral  of  3  x^  dx  (or  the  primitive  of  3  x^)  may  be 
x^  01  x^  -\-  $,  and  evidently  also  x^  -{-  ij  or  ^^  +  any  constant 

whatever.  In  general,  |  F\x)dx  is  F{x)  +  C,  where  C  is 
any  arbitrary  constant.  For  the  latter  expression  differenti- 
ated gives  the  former  (§  27). 

An  arbitrary  constant  (usually  denoted  by  C)  must  there- 


INTEGRAL    CALCULUS  59 

fore  always  be  supplied  after  integrating  any  differential  to 
obtain  the  complete  integral. 

78.  There  is  no  general  method  of  integration  known 
corresponding  to  the  general  method  of  differentiation  of 
Chapter  I.  The  only  way  we  arrive  at  the  primitive  of  a 
given  function  is  through  our  previous  knowledge  of  what 
function  differentiated  will  yield  the  given  function. 

79.  ^ax^dx  =  ^^  +  C, 
J  n  -\-  1 

ax^"^^ 
provided  n  is  not  =  —  i .     For  the  differential  of h  C 

n  -\-  1 

is  evidently  ax^dx  provided  «  -f  i  is  not  zero ;  i.e.  provided 
n  is  not  =  —  i. 

The  rule,  therefore,  for  integrating  the  simplest  algebraic 
function  is  to  increase  the  exponent  by  one,  and  divide  the 
coefficient  by  the  exponent  so  increased  (and  then,  of 
course,  to  add  an  arbitrary  constant). 

Thus,  (2x^dx  is  |  j«^  +  C. 

80.  Examples. 
2x  dx  =  7 


J" 

(zx^dx=l     Ans.  I^  +  C 

X 

Cdx^ 
J  x^ 

CAdx^ 
J    x^ 


X  dx  _  -, 
2 

x-^dx=l 


?       Ans.   -  -i-  4.  ^. 


60  INFINITESIMAL    CALCULUS 

8i.  It  may  seem  at  first  that  a  result  involving  an  arbitrary 
constant  can  be  of  little  use.  But  this  is  far  from  true. 
Though  we  cannot  determine  the  arbitrary  constant  from  the 
given  differentia],  we  may  have,  in  any  particular  problem, 
information  from  some  other  source  which  will  enable  us  to 
determine  it,  and  often,  as  we  shall  see,  we  do  not  need 
to  determine  it  at  all.  We  may  interpret  the  constant  C 
geometrically  by  plotting  the  equation  v  =  Fix)  -\-  C.  To 
know  F\x)dx  or  F\x)  is  to  know  the  slope  of  the  curve 
for  any  value  of  x.  But  evidently  the  slope  of  the  curve 
does  not  determine  the  curve ;  since,  if  the  curve  were 
shoved  up  or  down  without  change  of  form,  it  would  have 
just  the  same  slope  for  the  same  value  of  x.  The  constant 
C  has  to  do  with  the  vertical  position  of  the  curve.  It  has 
nothing  to  do  with  its  form. 

82.  We  may  profitably  follow  the  plan  adopted  in  intro- 
ducing the  differential  calculus,  and  begin  by  considering  a 
mechanical  and  a  geometrical  application. 

We  have  seen  that,  knowing  a  body  falls  according  to  the 
law  j=i6/2,  (i) 

we  can  show  that  its  velocity  at  any  point  is 

(is  .  /  V 

Suppose,  however,  we  only  know  that  a  body  acquired 
velocity  according  to  law  (2),  can  we  pass  back  to  law  (i)? 
As  has  been  said,  in  the  integral  calculus  it  is  customary  to 
use  the  differential  form  to  start  with.  Accordingly,  we 
write  (2)  in  the  form 

ds  =  32  /di. 
Integrating,  we  have 

s  =^32  ^d^  =  ^ -\-C  =  16  /'^  C.  (3) 


INTEGRAL    CALCULUS  61 

Now,  although  equation  (2)  with  which  we  started  does 
not  enable  us  to  judge  of  the  value  of  C,  we  may  evaluate  C 
from  outside  data. 

Thus  if  we  know  that  s  is  measured  from  the  point  at 
which  the  body  started  to  fall,  we  know  that  when  /  was  zero, 
s  must  have  been  zero  too. 

Putting  J-  =  o  and  /  =  o  in  (3),  we  have 

o  =  o  +  C, 

or  C  =  o. 

After  substituting  this  value  of  C  in  (3),  the  equation 
takes  the  definite  form 

J  =16  A 

83.  Of  course,  C  is  not  always  zero.  In  fact,  in  the  above  ex- 
ample, we  might  reckon  the  distance  s  of  the  falling  body  not  from 
the  point  where  it  started,  but  from  a  point  27  feet  above.  We  then 
know  that  when 

/  =  o,  J  =  27. 

Substituting  in  (3),  we  have 

27  =  o  +  C  or   C  =  27, 
and   (3)  now  becomes 

5=16/24-27. 

Evidently  the  value  of  C  depends  solely  on  what  origin  we  use  to 
measure  s  from. 

84.  Similarly,  if  we  know  the  relation  between  the  slope 

of  a. curve  ~  and  its  abscissa,  we  can  obtain  the  equation 
ax 

of  the  curve,  except  for  an  arbitrary  constant  which  regu- 
lates the  vertical  position  of  the  curve.  This  example  is  the 
true  inverse  of  the  geometrical  illustration  in  the  differential 
calculus  (§  12).  But  for  the  purpose  of  the  integral  calculus 
we  prefer  another  geometrical  example. 


62 


INFINITESIMAL   CALCUL  US 


85.  Suppose  we  have  (Fig.  10)  a  plot  oi  y=f{pc).  Give 
to  X  an  increment  Ajc,  viz.  AE  or  BK^  and  consider  the 
resulting  increment  not  oi  y^  but  of  the  area  OABC  or  z. 


This  increment  ^z  of  the  area  is  evidently  the  small  area 
ABDE.  This  small  area  is  the  sum  of  the  rectangle 
ABKE  and  the  tiny  triangle  BDK.  The  area  of  the  rec- 
tangle is  the  product  of  its  base  Ajc  by  its  altitude  fipc). 

So  that 

^z=/(x)^x-{-BnK.  (i) 

Evidently  the  smaller  we  make  Ajc,  the  smaller  the  area 
of  BDK  becomes  relatively  to  the  small  rectangle,  and  may 
finally  be  neglected,  giving  the  important  equation 

dz  —f{x)dx.  (2) 

This  is  not,  of  course,  a  mere  approximation.  It  is  abso- 
lutely exact. 


INTEGRAL    CALCULUS  63 

The  reasoning  just  given  is  to  be  understood  as  an  elliptical  form 
of  the  following: 

Dividing,  (i)  by  A^r,  we  have 


£=/^^)  +  ^-  (3) 


T..       BDK .   ,       ,, 
Now IS  less  than 

rect  HK ,  .^   rect  HK 
Ax      '  ^*^'      BLT 

But  the  area  of  a  rectangle  divided  by  its  base  is  its  altitude  —  in 
this  case  DJiT.     Hence  (3)  may  be  written 

Az 

—  =/(x)+  something  less  than  DIT. 

Ax 

It  is  evident  that  when  Ax  becomes  zero,  DJ^  becomes  zero,  and 
*'  something  less  than  DJi^  becomes  zero,"  so  that  our  equation  becomes 

ax 
which  may  be  written 

dz  —  f{x)dx. 

This  equation  is  often  written 

dz  =y  dXj  or  z  =  \y  dXy 

y  being  the  usual  symbol  for  f{x),  the  ordinate  of  a  curve. 
86.    Suppose  y  or  f{x)  to  be 

that  is,  \et  y  —  2,  x^  -\-  S  ^^  ^^^  equation  of  a  curve.  The 
integral  calculus  enables  us  to  obtain  the  area  z  in  terms  of 
the  abscissa  x. 

We  know  that         dz  =  (3  ^^  _|_  ^^  ^^^ 

z  =  jc-^  +  5  jc  +  C.  (i) 


64  INFINITESIMAL    CALCULUS 

The  student  may  test  the  correctness  of  this  integral  by 
differentiating  it  and  obtaining  (3^^  -f  ^dx. 

It  remains  to  determine  C.  Since  we  intended  to  meas- 
ure the  area  z  from  the  j-axis,  evidently  z  vanishes  when  x 
vanishes.  Putting  x  and  z  both  equal  to  zero  in  (i),  we 
obtain  C  =  o.  (If  we  had  measured  area  from  some  other 
vertical  than  the  jj^-axis,  the  value  of  C  would  be  different.) 
Hence  (i)  becomes  z=^  x?  -\-  ^x. 

Thus  suppose  x  —  y^  then  2  =  42.  That  is,  the  area  included 
between  the  curve  y  =  -^x^  ■\-  <^^  the  axes  of  coordinates  and  a  vertical 
3  units  from  the  ^-axis  is  42  units.  If  the  linear  units  be  inches,  the 
area  units  are  square  inches. 

87«  We  see  more  clearly  now  than  in  §  76  why  integration  was  first 
conceived  of  as  summation.  The  area  2  is  evidently  the  sum  of  a  great 
many  A2's,  and  at  the  limit  is  conceived  of  as  the  sum  of  an  indefinite 
number  of  dz^%. 

The  dz  is  thought  of  as  an  elementary  strip  of  area  infinitely  narrow 
—the  limit  of  ABDE, 

88.  The  problem  of  obtaining  curvilinear  areas  was  one  of  the 
earliest  and  is  one  of  the  most  important  of  the  applications  of  the 
integral  calculus.  Previous  to  the  discovery  of  this  branch  of  mathe- 
matics only  a  very  few  curves,  such  as  the  circle  and  parabola,  could 
be  so  treated. 

89.  We  are  here  chiefly  interested  in  the  geometrical 
symboUsm.  We  have  seen  that  the  slope  of  a  curve  is 
the  differential  quotient  of  its  ordinate  (with  respect  to  its 
abscissa).  We  now  see  that  the  ordinate  in  turn  is  the 
differential  quotient  of  its  area  (also  with  respect  to  the 
abscissa).     For  dz=ydx  means  simply 

dx 


INTEGRAL    CALCULUS  65 

If  we  wish  to  make  a  graphic  picture  of  any  function  and 
its  derivative,  we  can  represent  the  function  either  by  the 
ordinate  ^  of  a  curve  or  by  its  area,  while  its  derivative  will 
then  be  represented  by  its  slope  or  ordinate  respectively. 

If  we  are  most  interested  in  the  function^  we  usually 
employ  the  former  method  (in  which  the  ordinate  repre- 
sents the  function)  ;  if  in  its  derivative^  the  latter  (in  which 
the  ordinate  represents  the  derivative).  That  is,  we  usually 
like  t  J  use  the  ordinate  to  represent  the  main  variable  under 
consiileration. 

Jevons  in  his  Theory  of  Political  Economy  used  the 
abscissa  x  to  represent  commodity,  and  the  area  z  to  repre- 
sent its  total  utility,  so  that  its  ordinate  y  represented 
"  marginal  utihty "  {i.e.  the  differential  quotient  of  total 
utiHty  with  reference  to  commodity).  Auspitz  and  Lieben, 
on  the  other  hand,  in  their  Untersuchungen  Uber  die  Theorie 
des  Freises,  represent  total  utility  by  the  ordinate  and  margi- 
nal utility  by  the  slope  of  their  curve. 

90.  The  method  of  integration  enables  us  not  only  to 
obtain  the  particular  curvilinear  area  described,  but  also  an 
area  between  two  Hmits,  as  AB  and  A'B^  (Fig.  10).  Evi- 
dently this   area   is  the  difference  of  two  areas   OA'B'C 

and    OABC.     The   first  is   the   value  of    \  f{x)dx,  when 

OA^  (or  JC2)  is  put  for  x  in  the  integral  when  found,  while 
the  second  is  the  value  of  the  same  integral  for  x  =  OA 
(or  x^.     This  is  expressed  as  follows : 


and  is  called  an  integral  between  limits,  or  a  definite  integral 
The  reason  it  is  called  definite  is  that  it  contains  no  arbi- 


66  INFINITESIMAL    CALCULUS 

trary  constant,  for  this  constant  disappears  when  one  of  the 
two  integrals  concerned  is  subtracted  from  the  other. 


Thus,  if  C/(x)dx  be  J^(x)-\-C, 

f{pc)dx 


s 


means  simply      (^^(^2)  +  <^)  -  (^(-^i)  +  C), 

which  reduces  to  F{x^—F{x^,  for  C  must  be  taken  to  be 

the  same  in  both  integrals. 

The  area  between  the  curve  3jr2  +  5,  the  ;r  axis,  and  the  two  verti- 
cals erected  at  :r  =  2  and  ;f  =  4  is 

J^'CS-^^  +  5)0'^  =  [^3  +  5^  +  C]^4_[^  +  5  •*•  +  C]x=2  =  66, 

for  the  C  drops  out,  since  for  each  expression  the  area  is  measured  from 
the  same  vertical,  though  no  matter  what  vertical. 

It  is  usual  to  abbreviate  the  expression  for  limits. 

Xaf=4  /»4 

f(x)dXf  we  write   j    f{x)dx. 

91^  There  are  certain  general  theorems  of  integration 
corresponding  to  the  general  theorems  of  differentiation  of 
Chapter  II.     Of  these  the  two  most  important  are  : 

CKf(x)dx  =  K  C/{x)dx 

and  f[./i(x)±/2(x)±  "')yx 

=f/i(xyx±f/2(x)dx±j*/s(x)dx  ±.... 

The  proof  of  the  first  is  simple,  for  the  integral  of  the 
right  side  of  the  proposed  equation  is  X(F(x) -{-€),  or 
KF{x)-\-KC  or  KF{x)-\-C\  where  F{x)  means  the  primi- 


INTEGRAL    CALCULUS  67 

tive  of  f{x)  and  C  is  an  arbitrary  constant.  But  C  might 
as  well  be  written  C,  since  its  value  is  anything  we  please. 

The  integral  on  the  left  is  also  KFix)  +  C ;  for  this 
differentiated  gives  Kf{x)dx. 

The  proof  of  the  second  is  also  simple.  If  we  denote 
the  primitives  of /i(jc:),  /2(^),  •••,  by  Fi(x),  ^^(x),  •••,  it  is 
evident  that  the  integral  on  the  right  is 

F,(x)  +  C,  ±  F,{x)-\-C2  ±  F,{x)-hCs  ±  -, 
or  F,(x)±F^x)±  '-•  -\-C,  (i) 

where  C  is  Ci  +  C2  +  Q,  and  is  therefore  arbitrary.  The 
integral  on  the  left  is  the  same  quantity  (i),  for  the  differ- 
ential of  (i)  is  (§  26), 

d(F,(x)  ±  F,{x)  . . .  +  C)  =  dF,{x)  ±  dFlx)  •  •  • 
=f\{x)dx  ±f2(x)dx  •••  =(/i(^)  ±fj^x)  •••  )dx. 

92.    Examples. 

1.  Integrate  (i  -I-  a  +  b)x^  dx, 

2.  IntegrSitG  X- dx -{- y  x^  dx -{■  ^  x^  dx. 

3.  Integrate  {k  +  2){cx^  dx  +  ^x^  dx}. 

Ans.  (h+2)\-x^  +  -xT  +  c\^ 
'5  7  ' 

4.  If  the  velocity  of  a  body  increases  with  the  time  according  to  the 

formula  —  =  3  /2,  find  the  formula  for  the  distance  traversed. 

5.  How  far  does  it  move  between  the  instant  when  /  is  3  seconds 
and  that  when  /  is  5  seconds? 

6.  Find  the  expression  for  the  area  (corresponding  to  z  in  Fig.  10) 

for  the  curve  whose  equation  is  jj'  =  5  ^r-^  +  2.     Ans.  ^y—  +  2x  -^r  C. 

t.   What  is  the  value  of  that  area  for  the  point  where  ;r  is  I? 
Where  ;r  is  3?     Where  j  is  22? 

8.  What  is  the  area  between  the  curve,  the  jc-axis,  and  the  two 
verticals  erected  at  ;r  =  2  and  x  =  4?     Ans.  i(X). 

9.  Solve  the  same  problems  for  the  c\iT\e  jf  =  x^  +  14;    for  ^  = 
x^;   fory^  =  4rt';f. 

10.   Find  the  area  z,  for  y  =  a^';  y  =  log  (^  +  5) ;  y  =  sin x. 

Ans.  ^-^  +  C;  (^+ 5)  log(.;t:+ 5)  -  jr  +  C;  -cosx  +  C, 


68  INFINITESIMAL    CALCULUS 

93.    Just  as  we  may  differentiate  successively,  so  we  may 
integrate  successively. 

If  we  perform  the  integration 

I  f{x)dx  and  obtain  fxipc), 

we  may  then  take 

I  f\{pc)dx  and  obtain  y^(jp), 

and  then  j  f^{x)dx  and  obtain  fj^x^, 

etc.  etc. 

Instead  of  writing  |  f^{x)dx,  we  may  substitute  for  /^(x) 
its  value   |  f{x)dxy  and  we  shall  have 

which,  however,  is  usually  abbreviated  to   j   j  f(x)dx  dx,  or 
even  to  J    i  /{x)dx^. 

Similarly,  we  may  write 

I   I   \/(x)dx  dxdx,  or  I   |   l/(x)dx^,  etc. 

We  may  express  the  double,  triple,  etc.,  definite  integrals 
also.     The  full  form  for  the  double  definite  integral  would  be 

x=o    \_yx=h  _J 

which,  however,  may  be  condensed  to 


INTEGRAL    CALCULUS  69 


94.    To  apply  these  ideas  we  recur  to  our  old  example  of  a  falling 
body.     Suppose  our  first  knowledge  is  not  j  =  16/2  nor  —  =  32/,  but 

— —  =  32;  that  is,  we  simply  know  that  the  acceleration  is  a  given  con- 
dt'^ 

stant  (32  velos  per  sec),  or  to  be  more  general  let  us  call  this  con- 
stant^. d[—\ 

The  given  equation,  — ^  =  g,  means,  as  we  know,  -J^ — L  =  ^^  or 


(i) 


■gdt, 


whence,  integrating,  — =gt-\-C;  (l) 

di 

but  this  may  be  written  ds  =  gt  dt  -\-  Cdtj 

whence,  integrating  again,         s  =  \gt'^  -f  Ci  -\-  K,  (2) 

We  have  still  to  determine  the  arbitrary  constants  C  and  K.     If  the 

distance  s  is  measured  from  the  starting-point,  then    s  and  t  vanish 

simultaneously.     Substituting  zero  for  them  both  in  (2),  we  obtain 

It  remains  to  determine  C. 

To  do  this  we  take  equation  (i)  and  suppose  the  body  falls,  not 
from  rest,  but  with  an  initial  velocity  of  u  feet  per  second;  then  when 

ds  . 
t  IS  zero,  —  is  «, 

dt 
and  (i)  then  reduces  to 

u  =  o  ■{■  C    or     C  =  M. 

Substituting  C  =  u  and  A'  =  o  in  equation  (2),  we  have 

s=lgfi  +  ut, 

the  general  equation  of  falling  bodies. 

95*    The  process  which  we  have  followed  out  in  detail  from  the 

equation 

dh 

may  be  condensed  as  follows : 


70  INFINITESIMAL    CALCULUS 

90.    The  simple  transcendental  integrals  are  obtained  as  follows  : 
Since       d{€vi\x)=  zo'=>xdx^  then  \  0.0%  xdx  =  sinx  +  C. 
Since  a^(cos  jr)  =  —  sin  x  dx,  then  \  —  sm  x  dx  =  co6  ;r  +  C, 

whence  \  sin  (x^dx  =  —  cos;r  —  C  =  —  cos;«r  +  C, 

for  C  is  perfectly  arbitrary. 

Since       <aO  =  ^Itogo^^  ^j^^^  T^z^Log^^  =  a-  +  C, 
Log  e  J       Log  <f 

whence  f  ^x^^^  ^^L£gi  +  C*. 

J  Log  a 

Also/  Ca^dx=^+C. 

J  log  a 

Since  o^arc  sinx  = — -  ^      .,  then    j  — ::;3:^^:;:;  =  arc  sin;ir  +  C 

Vi  -  x^  -^  y/i-x'^ 

Since      d^arctan;r  =  — - — ,  then    \ =arctan:r+C 

I  +  x^  J  I  +  x^ 

Since  ^logjf=— ,  then    i  — =  logjf+C 

X  J    X 

=  ]ogx  -\-\ogir  z=\og{Jirx) 
lor  C  and  IT  are  wholly  arbitrary. 

97.   We  may  summarize  the  formulae  for  integration  which 
have  been  given : 

ax""  dx  = h  C  (when  n  is  not  =s  —  1), 

I  ax~^  dx  =  a  log  x  +  C*, 

J  Loga 


INTEGRAL    CALCULUS  71 


p*^^=^+  c, 


arc  tdiVix  +  C, 


^  Vi 


dx 


arc  sin  x  +  C, 


I  sin  jc  <'/.r  =  —  cos  x  -}-  C, 
I  cos  :r  //jc  =  sin  Ji?  +  C. 


98.  Treatises  on  the  integral  calculus  are  usually  ery  bulky,  be 
cause  they  are  occupied  with  the  determination  of  special  integrals, 
both  definite  and  indefinite,  and  with  special  devices  for  obtaining 
them.  In  this  little  book,  which  is  devoted  to  only  the  most  general 
and  fundamental  principles,  we  may  fitly  close  our  discussion  at  this 
point.  Practically,  even  advanced  students  of  the  Calculus  usually 
depend  on  tables  of  integrals.  The  reader  is  referred  to  B.  O.  Pierce's 
"  Short  Table  of  Integrals."  Completer  tables  occupy  large  quarto 
volumes.  An  absolutely  complete  table  does  not  exist,  for  there  are 
multitudes  of  integrals  which  have  never  yet  been  solved. 

99.  We  may,  however,  point  out  one  tool  for  integrating 
already  in  the  reader's  possession. 

Suppose  we  have  to  integrate 

X  {x^  H-  2) V.r. 
This  may  evidently  be  put  in  the  form 

(x^  -\-  2yx  dx, 

or  ^  (x^  +  2y  2  X  dXf 

or  i(x'+2yd(x^, 

or  i(^+2)V(jt:2+2), 

and  in  this  form  it  is  easily  integrated. 


72  INFINITESIMAL    CALCULUS 

For,  putting  u  =  x^  -{-  2,  we  have 

the  integral  of  which  is 


\u^du, 


or 


^+c, 


W  +  i)-^  +C. 


This  device  consists  in  changing  the  variable,  getting  rid 
of  dx,  and  obtaining  instead  a  differential  of  some  other 
variable,  u,  in  terms  of  which  the  whole  expression  may  be 
written. 

100.    Examples. 
1.    f.U  =  ?  6.    f^^^. 


X 


^     ^  2bx  dx 
8. 


Ans.  a^x-\-3a^x^+-^x^+'^x'' 
—  7.dx 


"•  j  Vf^' 


APPENDIX  73 


APPENDIX 

FUNCTIONS   OF   MORE   THAN   ONE   VARIABLE 

lOi.  We  have  had  to  do  hitherto  with  functions  of  only 
one  variable,  such  as  ;t:^  +  2  ^  -f  3.  But  the  magnitude 
x^^-\-  2  xy  -\-  ^  y,  for  instance,  is  dependent  for  its  value  on 
fwo  variables,  x  and  y ;  i.e.  is  a  function  of  x  and  y. 

The  relation  z  =  x^  -\-  2xy  -^  ^y,  or,  more  generally, 
z  =  F{x^  y)y  states  that  z  is  a  function  of  x  and  y ;  that  is, 
that  a  change  either  in  ^  or  j  produces  a  change  in  z. 

Thus,  the  speed  of  a  sailing  vessel  is  a  function  of  the  strength  of 
the  wind  and  the  angle  at  which  she  sails  to  the  wind. 

The  force  which  produces  tides  is  5.  function  of  the  earth's  distance 
from  the  moon  and  its  distance  from  the  sun. 

The  price  of  stocks  is  a  function  of  the  rate  of  dividends  and  of  the 
rate  of  interest. 

Similarly,  w  =  F{Xy  y,  z)  expresses  the  fact  that  iv  de- 
pends on  x^  J,  and  s,  and  so  on  for  any  number  of  variables. 

Thus,  the  force  which  guides  the  moon  is  a  function  of  its  distance 
from  the  earth,  its  distance  from  the  sun,  and  the  angle  between  the 
directions  of  these  two  distances. 

The  price  of  a  Turkish  rug  is  a  function  of  the  prices  of  its  constitu- 
ents, the  cost  of  transportation,  the  rate  of  tariff,  etc. 

If  for  w  =  F(x,  y,  z),  the  condition  of  some  special 
problem  should  require  z  to  remain  constant,  the  function 
may  be  written  as  w  =  (t>(x,  y)  ;  and  if  y  is  also  constant,  as 


74  INFINITESIMAL    CALCULUS 

Thus,  the  speed  of  a  sailing  vessel  is  a  function  of  her  angle  to  the 
wind,  if  the  strength  of  the  wind  remain  constant. 

The  price  of  woollen  cloth  is  a  function  of  the  price  of  wool,  if  the 
cost  of  labor,  etc.,  remain  constant. 

102.  Since  the  terms  of  an  equation  can  be  transposed, 
it  is  always  possible  to  gather  them  all  on  the  left  side,  thus 
reducing  the  right  side  to  zero,  y  =  V^^  +  i  is  the  same 
equation  as  ^  —  .r*^  —  i  =  o.  The  left  member  is  here  a 
function  of  x  and  y.  And  in  general  it  is  evident  that  any 
relation  between  two  variables  y  =  F{x)  can  be  reduced  to 
the  form  <^(x,  y)  =  o.  When  expressed  in  the  first  form,  y 
is  called  an  explicit  function  of  x.  In  the  latter  it  is  an 
implicit  function  of  x. 

In  like  manner,  any  relation  z=F(x,  y)  can  be  reduced 
to  the  form  <^(^,  y,  z)=o;  any  relation  w  =  F{x,  y,  z)  to 
<f>(x,  y,  z,  w)  —  o,  and  so  on. 

103.  We  have  seen  that  ^{x,y)=o  or  y=F{pc)  can 
always  be  represented  by  a  curve  with  x  and  y  as  the  two 
coordinates.  So,  also,  <^{x,  y,  z)=o  or  z  =  F(x,  y)  can 
always  be  represented  by  a  surface  with  x,  y,  and  z  as  the 
t/iree  coordinates. 

Draw  three  axes  at  right  angles  to  each  other,  such  as  the 
three  edges  of  a  room,  meeting  at  a  corner  on  the  floor,  the 
^-axis  being  directed,  say,  easterly,  thejj'-axis  northerly,  and 
the  2-axis  upward. 

To  represent         z  =  x^  -^  2xy  -\-  3  j-, 

let  X  have  any  particular  value,  such  as  2,  and  y,  i. 

Then         2=2^+2X2Xi-f3Xi"=ii. 

Find  the  point  in  the  room  which  is  2  units  east  of  the 
corner,  i  unit  north  of  it,  and  11  units  above  it.     This  is 


APPENDIX  75 

one  point  of  the  required  surface.  By  taking  all  possible 
combinations  of  values  of  x  and  y,  and  finding  the  result- 
ing values  of  s,  we  can  find  all  points  on  the  surface. 

104.  When  z  =  F(x,  y),  we  may  vary  x  by  Ax,  while  y 
remains  constant,  and  thus  cause  in  z  an  increment  denoted 
by  A2.     The  ultimate  ratio  of  Az  to  Ax  is  expressed  by 

bz        dF{x,y) 
dx  dx 

and  is  called  the  partial  derivative  of  F{x,y)  with  respect 
to  JC. 

Similarly,  dz^^dF{x,y) 

dy  oy 

is  the  partial  derivative  with  respect  to  y ;  i.e.  the  derivative 
obtained  by  keeping  x  constant  during  the  differentiation. 

Observe  that  the  symbol  d,  denoting  partial  differentia- 
tion, is  not  identical  with  d. 


105.  The  geometrical  interpretation  of  these  partial  deriv- 
atives can  be  made  evident.  If  on  the  surface,  z=F{x,y), 
say  the  surface  of  a  stiff  felt  hat,  we  take  any  given  point  F 
and  pass  through  it  a  vertical  east  and  west  plane,  the  plane 
and  surface  intersect  in  a  curve  passing  through  F.  The 
tangential  slope  of  this  curve  at  F  (or,  as  we  may  call  it,  the 

dz 
E-W  slope  of  the  surface  itself)  is  —  •  For  the  coordi- 
nates of  F  are  x,  y,  z,  and  those  of  a  neighboring  point  Q 
on  the  curve  (and  therefore  on  the  surface)  are  x  -f  Ax,  y, 
z  -f  Az,  where  A^  is  the  difference  between  the  ^'s  of  F 
and  Q,  and  Az  the  difference  between  the  z's  ;  the  jf's  are  by 
hypothesis  the  same.    The  slope  of  the  line  joining  /*and  Q  is 


76  INFINITESIMAL    CALCULUS 

— ,  and  its  limiting  value,  lim  —  or  — ,  is  the  slope  of 

A^  A^         ox 

the  curve  at  P  (see  §  12);  i.e.  the  E-W  slope  of  the  sur- 
face. 

Similarly,  -^,  or  — V^-^,  is  the  north  and  south  slope  of 
by  ay 

the  surface. 

These  two  primary  slopes  of  the  surface  can  be  repre- 
sented by  placing  two  straight  wires  or  knitting  needles 
tangent  to  the  hat  at  the  point  P,  one  in  an  E-W  vertical 
plane  and  the  other  in  a  N-S  vertical  plane. 

If  we  take  any  neighboring  point  R  on  the  surface,  its 

coordinates  are  x  +  t^x,  y  +  Aj',  z  +  As,  where  the  A's  are 

the  differences  of  coordinates  of  P  and  R. 

Az 
Join  P  and  R.     Then  —  represents,  not  the  true  slope 

Ax 

of  the  line  PR,  but  its  easl  and  wes/ slope  (not,  of  course,  the 
east  and  west  slope  of  the  surface  itself).  It  is  the  rate  the 
line  ascends  in  comparison,  not  with  its  true  horizontal  prog- 
ress, but  with  its  eastward  progress.  A  climber  ascending  a 
northeasterly  ridge  may  be  rising  5  feet  for  every  3  of  hori- 
zontal progress,  but  yet  rising  5  feet  for  every  2  of  eastward 
progress.     We  have  to  do  with  the  latter  rate,  not  the  former. 

So  also  —  is  the  north  and  south  slope  of  the  same  line  PR. 
Ay 

Now  let  R  approach  P  (along  any  route  whatever  upon 

the  surface)  until  it  coincides.     The  line  PR  approaches  a 

hmiting  position  which  is  a  new  tangent  to  the  surface  (a 

tangent  to  that  curve  in  the  surface  which  R  traced  in  ap- 

Az 
proaching  P).     The  E-W  slope  of  this  tangent  is  lim  -— , 

called  — ,  and  its  N-S  slope,  --- 
dx  dy 

Representing  this  tangent  by  a  third  wire,  we  have  three 


APPENDIX  77 

tangent  wires  through  P,  one  in  an  E-W  vertical  plane,  a 
second  in  a  N-S  vertical  plane,  and  the  third,  any  other 
tangent.     The  first  has  no  N-S  slope ;    its  E-W  slope  is 

— .     The  second  has  no  E-W  slope ;  its  N-S  slope  is  —  • 

^•^  dz     dz     ^y 

The  third  has  both  kinds  of  slope,  viz.,  —  and  — 

dx  dy 

1 06.  As  will  be  shown,  the  relation  between  these  various 
derivatives  is 

dz  =  -^dx-\--^dy,  (i) 

ox  dy 

which  mav  be  thrown  into  the  forms : 

dz  _  dz  .   dz  dy 

dx      dx  dy  dx 

or  [ .  (2) 

dz__dz_  dx  dz 

dy      dx  dy  dy . 

The  form  (i)  has  the  great  advantage  of  symmetry.     It 

seems,  however,  to  conceal  the  existence  of  ^  or  — ,  which 

dx       dy 

are  brought  out  in  (2).     These  last  two  magnitudes  require 

merely  a  word  of  explanation.     -^  is  not  an  upward  slope 

dx 

at  all,  as  it  does  not  involve  the  vertical  z.  It  is  the  incli- 
nation of  the  third  wire  across  the  floor,  the  rate  at  which 
a  moving  point  on  it  proceeds  north  in  relation  to  its  east- 
ward progress. 

107.  The  proof  of  the  formula  stated  in  the  last  section  is  as 
follows :  * 

*  In  order  to  master  and  remember  this  proof,  the  student  is  advised 
to  construct  for  it  some  actual  physical  model.  He  will  then  find  it 
extremely  simple. 


78  INFINITESIMAL    CALCULUS 

We  first  assume  that  all  wires  through  P  tangent  to  the  surface  lie 
in  one  and  the  same  plane  called  the  tangent  plane.  This  assumption 
is  analogous  to  that  in  §  14,  that  the  progressive  and  regressive  tan- 
gents coincide.  There  is  an  exception  if  the  surface  has  an  edge  or 
vi'rinkle  at  the  given  point. 

Let  us  take  in  this  plane  the  three  tangent  wires  above  considered, 
viz.  the  two  primary  wires  (in  vertical  planes  running  E-W  and  N-S 
respectively)  and  the  wire  obtained  as  the  limiting  position  of  Z*^. 
Take  a  point  Q'  on  this  third  or  "  general "  wire,  having  coordinates 
X  +  A'jf,  y  +  cJy,  z  +  tJz.  (The  primes  serve  to  distinguish  Q^  on 
the  tangent  plane  from  Q  on  the  surface.) 

Through  Q^  pass  two  vertical  planes  running  E-W  and  N-S  respec- 
tively. We  already  have  two  such  planes  through  P.  These  four 
vertical  planes  cut  the  tangent  plane  in  a  parallelogram,  of  which  PQ 
is  a  diagonal  and  the  "  primary  wires  "  are  the  two  sides  meeting  at 
P.  Denote  the  two  vertices  as  yet  unlettered  by  H  and  K,  the  former 
being  in  the  E-W  and  the  latter  in  the  N-S  primary  wire. 

A'z  being  the  difference  in  level  of  P  and  Q^  is  the  sum  of  the  dif- 
ference in  level  of  P  and  H  and  of  H  and  Q\  just  as  the  difference  in 
level  between  Mount  Blanc  and  the  sea  is  the  sum  of  the  elevation 
of  Lake  Lucerne  above  the  sea  and  of  Mount  Blanc  above  the  Lake. 
(It  does  not  matter  whether  H  is  or  is  not  intermediate  in  level  between 
P  and  Q\  for  if  not,  one  of  the  heights  considered  becomes  negative.) 

Now  the  difference  in  level  of  P  and  H  is 

for  the  difference  of  level,  h,  between  any  two  points,  as  M  and  N 


(Fig.  11)  is  the  product  of  the  biope  of  MN  hy  the  horizontal  interval, 
a,  between  them  (since  :  slope  of  MN  —  - ,  whence  /i  =  a  X  slope  of 


a 


APPENDIX  79 

MN^.     ^  is  known  to  be  the  slope  of  PO ,  and  a';c  is  the  E-W 

interval  between  P  and  Q ,  and  therefore  also  the  E-W  interval  (or 
in  this  case  the  horizontal  interval)  between  /'and  H  (since  JI B.nd  Q 
are  in  the  same  N-S  plane). 

Again  the  difference  in  level  between  ZTand  Q  is 

by 
For  — ,  being  the  slope  oi  PK,  is  also  the  slope  of  HQ'  parallel  to 

dy 

PK,  and  A'y,  being  the  N-S  interval  between  P  and  Q',  is  also  the 
N-S  (and  in  this  case  horizontal)  interval  between  H  and  Q'  (since 
H  and  P  are  in  the  same  E-W  plane). 
Therefore, 

A'2=-^A';r  +  ^A'>/.  (I)' 

dx  By 

which  is  the  prototype  of  the  desired  result  (l). 

_,  .  ,  A's      dz      dz      A'y  (2)' 

This  may  be  written     —7—  =  ^ — h  -tt-  *  tt~' 

^  A'x     (jx     dy    A'x 

Now  — -    is  the  E-W  slope  of  the  "general  tangent"  wire  PQ'. 

But  we  have  seen  that  —  is  also  this  slope.   Again,  —^  is  the  inclina- 

dx  A'x 

tion  of  this  same  wire  across  the  floor  (the  rate  at  which  a  point 
moving  on   the  wire  proceeds  northward  relatively  to  its  eastward 

progress).     But  so  also  is  -^  (§  io6)-     Substituting  therefore  these 

dx 
values  for  the  primed  expressions,  we  have 


dz  _  Sz  ■   dz     dy^ 
dx     Qx      dy    dx 


which  may  be  thrown  into  the  form 
^d: 

dx  By 


dz  =  ^dx  +  ^dy. 


In  this,  dz  is  called  the  total  differential  of  2,  while  ^  dx  and  ^  c^ 
are  \\s  partial  differentials. 

It  is  evident  that  we  should  reach  the  same  result  if  in  the  preced- 
ing reasoning  we  had  employed  K  in  the  way  we  did  employ  H,  and 


80  INFINITESIMAL    CALCULUS 

vice  versa;  also  that  we  could  have  divided  (i)'  by  A'_y  instead  of 
by  tJx. 

1 08.  The  formula  (i)  (§  106),  or  its  two  alternative 
forms  (2),  enable  us  to  ascertain  the  direction  of  any  tan- 
gent line  to  a  surface. 

Thus,  let  the  surface  be 

0  =  ;ir2  +  2  AT^  +  3  /2, 

and  let  it  be  required  to  determine  any  tangent  line  at  the  point  whose 
X  and  y  are  i  and  i  respectively;  z  is  evidently  6. 

1.  The  primary  E-W  tangent  wire  at  this  point  has  an  E-W  slope 
^=2;f+2jj/  =  4,  found  by  differentiating  the  above  equation  treat- 
ing  y  as  constant,  and  has  no  N-S  slope. 

2.  The  primary  N-S  tangent  wire  at  this  point  has  a  N-S  slope 

^=2x-^6y  =  S,  and  has  no  E-W  slope. 
dy 

3.  The  tangent  wire  in  the  vertical  plane  running  northeast  and 
southwest  has  an  E-W  slope  of 

(/x      dx      dy    dx 

=  4  +  8^  ■ 

dx 


and  a  N-S  slope  of 


=  4  +  8  X  I  =  12, 

dz  _Sz  _  dx  .  dz 
dy      dx     dy      dy 


=  4  X  I  +  8  =  12. 

4.  The  tangent  wire  in  the  vertical  plane  running  northwest  and 
southeast  has  the  two  slopes 

4+8(-i)  =  -4 
and  4(-i)+8=+4. 

5.  The  tangent  wire  in  the  vertical  plane  cutting  between  north  and 
east  so  as  to  be  advancing  north  twice  as  fast  as  east 


li.e.  so  that  ^=2), 


APPENDIX  81 

has  slopes  of  ^±  =  ^^^  .± 

ax      Qx      dy     dx 

=  4  +  8  X  2  =  20, 

and  ^  =  5i.^+5f 

dy      ^x     dy      Qy 

=  4  X  ^  +  8  =  10, 

and  so  on  for  any  tangent  wire  whatever. 

109.  Examples. 

1.  Find  the  slopes  of  the  five  sorts  above  indicated  for  the  same 
surface  at  the  point  for  which  ^  =  3  and  y  =  2. 

2.  At  the  point  where  ^  =  —  i,  y  =.—  i. 

3.  At  the  point  where  x  =  o,  y  ■=  o. 

4.  For  the  surface  z  =  x^-\-x'^-\-x-\-xy+y-\-y^-{-y^  at  the 
point  X  ^=^0,  y  =  1, 

5.  For  the  surface 

z  =  x'^y-  2  xY^  +  3 

at  the  point  x  =  2,  y  =  '^. 

6.  On  the  same  surface  at  the  same  point,  what  are  the  E-W  and 
N-S  slopes  of  the  tangent  line  which  progresses  northward  3  times 
as  fast  as  eastward  ?   4  times  ?    3I  times  ? 

7.  Answer  the  same  questions  for  2  =  log  jJ^  +  3*  +  xy. 

110.  When  we  have  a  function  of  more  than  two  vari- 
ables, as  w=F{x,  y,  z),  there  is  no  mode  of  geometrical 
interpretation  corresponding  to  the  curve  for  y  =  F{x)  and 
surface  for  z  =  F{x,  y)  (unless,  indeed,  we  posit  a  "  fourth 
dimension,"  and  speak  of  a  "  curved  space  "  of  three  dimen- 
sions whose  coordinates  are  x,y^  z,w\). 

It  may  be  shown,  however,  in  a  manner  strictly  analogous 
to  the  process  of  §  107,  but  without  employing  the  geomet- 
rical image,  that 

,         bw    ,     ,   bw    ,     ,  bw  J 

dw  =  ——  ax  H dy  -\ dz. 

ox  dy  dz 


82  INFINITESIMAL    CALCULUS 

This  differential  equation  is  elliptical  for  the  three  equation^i 
obtained  by  dividing  through  by  dx,  dy,  and  dz. 

The  theorem  and  its  proof  are  extensible  to  any  number 
of  variables. 

111.  A  very  important  application  of  the  principle  of 
partial  derivatives  occurs  when  we  have  but  two  varial^les, 
but  y  is  an  implicit  function  of  x  ;  i.e.  when  <^{x,  y)  —  o. 

We  are  enabled  to  obtain  the  derivative  --^-  without  being 

dx 

obliged   first   to    transform  the  implicit  function  into  the 
explicit  form  y^^F{x), 

Thus,  if  x^  -^  y^  =  25,  we  may  find  -^  without  changing  the  equa- 
tion to  the  form  >'  =  ±  V25  —  x^. 

112.  We  know  from  §  106  (2)  that  if  z  =  <^{x,y)^  then 

dz_  ^  d<f>{x,y)      d<f>{x,y)  ^  dy^^ 
dx  dx  dy  dx 

which  may  also  be  written  in  two  other  forms,  as  given  in 
§  106. 

When  z  is  zero,  as  in  the  case  now  being  considered,  then 

—  is  also  zero  (§  27,  end).     Making  this  substitution  in  the 

dx 

above  equation,  we  obtain 

dy  _  dx 

dx~      d<t>(Xfyy 
dy 

In  words  :  To  find  the  differential  quotient  of  y  with  re- 
spect to  X  when  the  functional  dependence  between  x  and y  is 
expressed  in  the  implicit  form  <f)(x,y)  =  o,  differentiate  the 
function  (fy{x,  y)  with  respect  to  x,  treating  y  as  constant, 
and  therr  again  with  respect  to  y,  treating  x  as  constant. 


APPENDIX  83 

Take  the  partial  derivative  found  from  the  first  differentia- 
tion^ divide  it  by  that  found  from  the  second,  and  prefix  the 
minus  sign. 

Thus,  if  x^  +  ^2  — .  2^,  or  x^  -\-  y^  —  2^  =  o,  we  may  find  -^  as 
follows:  "^-^ 

The  partial  derivative  of  x^  -\-  y^  —  25  with  respect  to  ;f  is  2  x,  and 
with  respect  to  y,  2.y.     Hence 

dy  _  _  2^  —  _-^ 

dx  2  y         y 

This  result  is  expressed  in  terms  of  both  x  and  y^  but  ic  may  be 
transformed  so  as  to  involve  but  one  variable.  Thus,  substitute  for  j^'  its 
value  as  obtained  from  x'^  ■\-  y^  —  25,  viz.  ±  V25  —  x'^.    Then 

dy X 

dx  ±  ^25  -  x"- 

a  result  identical  -vs  ith  that  obtained  by  differentiating  the  explicit  form 


J  =  ±  V2J 


113.    Examples. 


1.  Find  !^,  if  xy  =  i. 

dx 

2.  Find  ^,  if  2  ;r2  +  3  ;/2  _  4  =  q. 

dx 

3.  Find  ^,  if  axY  +  ^^Y  =  O- 

4.  Find^,  if^-±i^  +  ^  +  ^=o. 

dx        X  —y      cy      k 


5.    Find  -^,  if  zo<s,{xy)=.x 
dx 


dy 
dx 

6.  Find  ^,  if  log(.r272)  +  ^^3  +  y?  +  2  ^t^/  +  a  =  o. 

dx 

7.  Show  §  1 1 2  geometrically. 


8+  INFINITESIMAL    CALCULUS 

114.  Functions  of  many  variables  are  peculiarly  appli- 
cable in  economic  theory,  though  as  yet  they  have  been 
very  little  employed.*  Many  fallacies  have  been  committed 
from  lack  of  this  more  general  conception  of  functional  de- 
pendence, and  from  the  tacit  assumption  that  mere  curves 
are  capable  of  delineating  any  sort  of  quantitative  relation. 
This  is  an  error  only  one  degree  less  flagrant  than  the  errors 
of  those  whose  sole  mathematical  idea  is  that  of  the  con- 
stant quantity. 

♦See,  however,  Edgeworth's  Mathematical  Psychics,  1881;  the 
author's  Mathematical  Investigations  in  the  Theory  of  Value  and 
Prices,  1892;    and  Pareto's  Cours  d"* economic  politique,  1896-7. 


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